Everton playmaker James Rodriguez's moment of quality was cancelled out by an error from goalkeeper Jordan Pickford which allowed Leicester to leave Goodison Park with a 1-1 draw.
Carlo Ancelotti's side had soaked up a huge amount of pressure following the Colombia international's brilliant 30th-minute opener but a defence which had conceded only three goals in six matches were just about hanging on.
However, they were undone when the England number one, watched by Gareth Southgate, failed to keep out Youri Tielemans' low shot.
Pickford may argue he saw the ball late as it came through a crowd of players but he got both hands to it but only pushed it in via a post.
It was the least Leicester deserved after the interval in particular as the hosts struggled to get out of their own half, dropping deeper and inviting pressure on themselves.
Ancelotti's flexible formation had begun the game switching from 4-2-3-1 in possession to 4-4-2 without the ball but for much of the second half they resorted to a back-five in order to cope with the incessant waves of attacks.
The sight of Rodriguez, whose class on the ball had produced a superb opener with a rapier-like shot off the post, launching himself into the air to block a James Maddison cross highlighted just how much pressure Everton were under.
Ancelotti had said the next four days, with Newcastle to come at home this weekend, would have a huge bearing on the rest of their season in terms of whether they can challenge for the top four.
On this evidence they will have to be more progressive - they finished the match with just 35 per cent possession - if they are to fulfil their manager's Champions League dream even with two matches in hand over most of their rivals.
Leicester, however, remain in the mix as they maintained third place but they will be frustrated they could not consolidate their position.
Even without the injured Jamie Vardy their pressing game had put the home side on the back foot from the start but they failed to create a genuine chance despite Harvey Barnes causing problems cutting in from the left.
It was something of a surprise to see Everton take the lead in those circumstances but there was little surprise in the goalscorer.
Andre Gomes' raking crossfield diagonal was touched into the path of Rodriguez by Lucas Digne, playing further forward than normal on the left, and the Colombian's left-footed shot was blocked by Wesley Fofana.
The Leicester defender could not keep control and allowed it to run back to the former Real Madrid playmaker who, with the Foxes' defence slow to close him down, curled in his first goal since October off the upright.
It was his 19th career goal from outside the penalty area in Europe's top five leagues but the first with his right foot.
The loss of Wilfred Ndidi just before half-time, replaced by Nampalys Mendy, interrupted Leicester's rhythm but they quickly recovered it after the break.
A clever move from a short corner saw Barnes find space in the penalty area and his shot was parried by Pickford, who was rested for the weekend's FA Cup win over Sheffield Wednesday, with James Justin placing the follow-up wide from the edge of the area.
Pressure continued to build and it was Pickford who finally cracked when he failed to deal with Tielemans' low shot through a forest of legs.
Dominic Calvert-Lewin's one big chance, having been starved of service, came in the 90th minute but he headed Digne's cross wide.