Southampton moved out of the Premier League relegation zone with a 1-1 draw at Everton, who scored an equaliser in stoppage time, with Stoke earlier relegated - but West Brom fight on following a dramatic day at the bottom of the table.
The Potters' fate was confirmed after losing 2-1 at home against Crystal Palace in Saturday's early kick-off.
The Baggies then gave themselves hope of staying up after a stoppage-time goal from Jake Livermore grabbed a 1-0 home win over Tottenham, which moved them off the bottom.
Southampton, though, could have still sent them down with victory at Goodison Park, which looked a realistic prospect after winger Nathan Redmond headed in after 56 minutes.
Defender Maya Yoshida, though, was sent off for a second caution with five minutes left, and the Saints could not hold out as Tom Davies' deflected shot secured a point for Everton deep into stoppage time.
The point moves Southampton up into 17th on goal difference above Swansea, who lost 1-0 at Bournemouth earlier.
It also means Burnley will qualify for Europe, while West Ham, who earlier won 2-0 at Leicester, also cannot be relegated.
Stoke's 10-year stay in the Premier League, though, came to an end following a 2-1 defeat by Crystal Palace at the bet365 Stadium.
The Potters started the day in 19th place, three points adrift of safety heading into their penultimate game of the season.
Xherdan Shaqiri put the home side in front with a free-kick after 43 minutes of Saturday's lunchtime game.
However, the Eagles, looking to make sure of their own top-flight status, hit back through James McArthur after 68 minutes before Patrick van Aanholt scored a second with four minutes left to seal Stoke's fate.
West Brom had also seemed to be heading for the Sky Bet Championship before Livermore's stoppage-time winner against Spurs at The Hawthorns.
It was a third win under caretaker boss Darren Moore, which came when Livermore bundled the ball in to spark jubilant scenes from the home supporters.
Bournemouth, meanwhile, secured another campaign in the top flight with their win over Swansea.
Ryan Fraser's deflected effort broke the deadlock after 37 minutes after a well-worked free-kick, which proved enough to move the Cherries mathematically clear of any danger.
West Ham are on 38 points with two matches left after beating Leicester.
The visitors took the lead after 33 minutes when Austrian forward Marko Arnautovic set up Joao Mario.
Irons captain Mark Noble added a fine second after 64 minutes with a 20-yard volley to seal a crucial victory and heap more pressure on Leicester manager Claude Puel.
Watford will also be playing in the Premier League again next season after a 2-1 home win over Newcastle, which was a first victory in eight games.
The Hornets took an early lead when midfielder Roberto Pereyra slotted in after just two minutes.
Argentinian winger Pereyra then turned provider after 27 minutes as his cross was headed in by Andre Gray at the back post.
Captain Troy Deeney missed the chance to make it 3-0 just before the break when his penalty was saved by Martin Dubravka.
Newcastle, already secure of their own Premier League status, pulled a goal back after 55 minutes through Ayoze Perez.