Chelsea's march towards the league title continued at Stoke, but they had to rely on a Gary Cahill goal in the 87th minute to grab a 2-1 win.
Antonio's visitors went ahead in the 13th minute when Stoke goalkeeper Lee Grant could only palm Willian's free-kick inside his own near post.
Mark Hughes' men levelled through a Jonathan Walters penalty in the 38th minute, but they were denied a point when Cahill, who had conceded the spot-kick, made amends with a late winner which took the Blues 13 points clear at the top.
Worse was to come for Stoke, who had Phil Bardsley sent off for a second booking in added time.
Leicester continued their remarkable resurgence under Craig Shakespeare as a flying start set them on their way to a 3-2 win at West Ham.
Goals from Riyad Mahrez and Robert Huth had the Foxes two up after only seven minutes and, although Manuel Lanzini pulled one back in the 20th minute with a brilliant free-kick, a Jamie Vardy finish restored the visitors' two-goal cushion before the break.
Andre Ayew headed home a second for the Hammers, but they could not find a leveller.
Victory maintained Shakespeare's perfect record since replacing Claudio Ranieri and lifted his side six points clear of the bottom three.
Everton striker Romelu Lukaku ended the week how he had started it - in the headlines - with two goals in the 4-0 win over 10-man Hull.
On Tuesday the Belgium international told the club he would not be signing the new £140,000-a-week contract to make him the highest-paid player in their history as he questioned whether they had the ability to reach the Champions League.
His double on Saturday, with both goals in added time, may not have been the decisive factor in putting the Toffees level on points with fifth-placed Arsenal, but it at least showed his professionalism cannot be called into question.
The platform had been laid by a first goal for 20-year-old Dominic Calvert-Lewin, in his sixth Premier League appearance, and, after Tom Huddlestone's red card, another from Enner Valencia. Lukaku then took centre stage to register his 20th and 21st league goals of the season, the first time an Everton player has reached that tally since Gary Lineker in 1985-1986.
The win puts Everton tantalisingly close to the top four, five points behind Liverpool with the Merseyside derby at Anfield up next after the international break.
Lukaku's comments angered Koeman, but the Dutchman knew only too well dropping the Premier League's top-scorer would amount to cutting off his nose to spite his face.
However, for nearly 80 minutes an unconvincing performance against relegation strugglers appeared to be adding weight to the striker's argument.
Lukaku always had Curtis Davies or Harry Maguire - two of Hull's three centre-backs - in close attention and, apart from scuffing his first attempt and being flagged offside as he stroked a shot past Eldin Jakupovic, he was barely involved in the first half.
The Belgian picked up after the break, with one shot rifled over from the edge of the penalty area and another fierce angled drive tipped behind by Jakupovic, which had the Gwladys Street End singing his name.
He did, however, finally produce a significant contribution with a lobbed assist for Valencia to score Everton's 78th-minute second within a minute of coming off the bench before confidently beating Jakupovic twice in added time.
The manner in which one grateful fan gleefully accepted Lukaku's shirt when he hurled it into the crowd at the final whistle showed his harsh words have not affected his popularity - helped of course by his goals.
With all the focus on the club-record signing's future, it was a trio from the club's youth teams who claimed the spotlight with the eighth-minute opener.
Barkley's piercing through-ball inside wing-back Andrew Robertson picked out Tom Davies, timing his run perfectly, and his low cross was side-footed home by Calvert-Lewin, who has emerged from the under-23s this season after arriving from Sheffield United in 2016. It was his first start since early January after injury.
But the pace dropped when Gareth Barry replaced the injured Morgan Schneiderlin and Everton were almost caught napping when Ashley Williams' hesitation in dealing with a long ball saw Sam Clucas' lob land on the top of the net just before half-time.
Koeman switched formation to match Hull's three at the back as the visitors began to gain the upper hand, only for Huddlestone's dismissal for a shin-high challenge on Idrissa Gana Gueye to halt their momentum in the 73rd-minute.
It released the pressure building up on the hosts, who quickly added the second to finally kill off the game before Lukaku stole the spotlight once again.
Tom Heaton denied Sunderland a dramatic late winner as David Moyes' men were left mired in relegation trouble after spurning a series of chances against Burnley at the Stadium of Light.
Heaton saved well from Fabio Borini in the 87th minute while Billy Jones and Sebastian Larsson also missed fine opportunities as the Clarets recorded only their third away point of the season.
Burnley themselves will be disappointed not to go home with the points with Ashley Barnes prodding their best chance wide in the first period and substitute Sam Vokes coming close late on.
Crystal Palace moved closer to preserving their Premier League status by earning their third successive victory with 1-0 home win over Watford.
Troy Deeney's second-half own goal proved enough to secure all three points in an otherwise cagey affair that left them four clear of the bottom three and took them up to 16th.
Two recent wins had already significantly reduced the pressure that had built on Sam Allardyce's team, providing the momentum needed to eventually avoid relegation after a year-long decline.
Their manager had long spoken of the need to produce a run of positive results, but he has also frequently stressed he would accept draws amid occasional victories if they avoid defeat.
The way Palace started at Selhurst Park suggested he had told his team this was one of those occasions when a draw would suffice, but with Chelsea, Southampton, Arsenal and Leicester their next four fixtures they would have been taking a risk.
Allardyce made only one change to the team that won 2-0 at West Brom a fortnight ago, with Jeffrey Schlupp replacing the injured Patrick van Aanholt. The greater disruption may have come from them not playing since.
Watford's Daryl Janmaat and Craig Cathcart also came in for Stefano Okaka and Jose Holebas, but for the tedious opening 45 minutes there was little by way of a threat on goal.
Wilfried Zaha and Andros Townsend combined well to create space for the latter to cross to Mamadou Sakho in the 39th minute when Miguel Britos headed clear, shortly before Zaha struck over the crossbar. Then on the stroke of half-time, the visitors' Valon Behrami also volleyed high and wide from the edge of the area after the ball had fallen kindly to him.
Until Townsend's run down the right wing and cross in the 65th minute, neither Palace nor Watford came any closer. Younes Kaboul should have cleared the winger's delivery but it instead reached Zaha, who then created space amid pressure from Cathcart before again shooting harmlessly over.
It had by then become clear that one goal would likely secure victory for either team, and almost immediately afterwards Zaha took advantage of space on the left to make ground until being brought down by Britos.
From the resultant 69th-minute free-kick Yohan Cabaye sent an inswinging cross into the area, and at the near post - despite little pressure - Watford captain Deeney sent a powerful header beyond goalkeeper Heurelho Gomes' dive, beating him on his right.
Watford could no longer be content with the point that had been looking likely, but beyond a long-range strike from substitute Abdoulaye Doucoure that tested Wayne Hennessey, they offered little in response.
Benteke later hesitated when through on goal, allowing Kaboul to challenge him, and Zaha's pace on the left continued to create space, yet still a second score eluded the hosts.
In the end it mattered little. Victory took Palace to within three points of their 13th-placed visitors, who top the group at the bottom of the table that Allardyce has demanded Palace lead.
