Chris Wood, who was recently wanted by Wolves, earned Leicester a point four minutes from time after Steven Naismith had restored Everton’s advantage just before half-time.
Summer signing Leonardo Ulloa had earlier hauled Leicester back on level terms just two minutes after Everton had gone in front through Aiden McGeady’s cracker.
Everton took the lead on 20 minutes when Leighton Baines’ low shot from distance deflected into the path of Sylvain Distin, who was denied by Kasper Schmeichel, but the ball broke to McGeady to curl a stunning shot into the top corner.
Leicester equalised just two minutes later when Distin’s clearance from a corner struck Ulloa and the hosts’ record signing flashed his shot past Tim Howard.
But Everton regained the lead at the end of the first half when Baines’ cut-back was touched on by Steven Pienaar and Naismith swept a first-time shot in off the crossbar.
Last season’s Championship winners should have equalised when Jeffrey Schlupp was sent clean through, but the substitute wildly fired his shot from the edge of the area into the crowd.
Gareth Barry was fortunate not to be shown a second yellow card before Leicester equalised again with four minutes to go when Wood rolled a shot past Howard after the ball had broken kindly for the substitute.
Roberto Martinez's Everton team featured no new faces, although Romelu Lukaku and Gareth Barry both made their first appearances since joining permanently.
John Stones stepped in at right-back for Seamus Coleman, who was only fit enough for the bench, while new signings Muhamed Besic and Christian Atsu were both unused substitutes.
Ross Barkley's injury certainly cast a cloud over the start of Everton's season, the England midfielder sustaining medial knee ligament damage in training.
Nothing could dampen Leicester's enthusiasm for their first Premier League season in 10 years, although they too were hit by injuries, with Marc Albrighton, Jamie Vardy and Matthew Upson all sidelined.
The hosts more than held their own in an even start and Howard had two nervy moments, first flapping at an Anthony Knockaert cross and then hesitating over a long ball and almost allowing Ulloa in.
But it was Everton who opened the scoring in the 20th minute, exploiting a fortunate deflection to the full.
Baines' long-distance shot did not look set to pose any problems until it was half blocked on the edge of the area and fell for Distin 15 yards out.
The defender's shot was saved by Schmeichel, but the goalkeeper could only watch as McGeady's delightful effort from the left curled over him and into the top corner.
But back came Leicester and within two minutes they were level.
The Foxes had already demonstrated their threat from set-pieces and, when Everton failed to clear a corner, record signing Ulloa repaid some of his transfer fee with a well-struck shot from the middle of the box.
Two minutes later Everton thought they had gone back ahead when Baines set up Naismith, but the ball had clearly gone behind before the defender played it.
Baines was a real thorn in Leicester's side and he wanted a penalty after going down in the area under pressure from Knockaert, but referee Mike Jones decided any contact had been accidental.
Everton began to gradually assert their dominance as the half went on and Schmeichel was relieved to see shots from McGeady and Naismith both miss the target.
But with the 45 minutes almost up, Naismith put the visitors back in front. Baines and Pienaar helped the ball in from the left and Naismith wrapped his left foot around it, sending a shot crashing in off the bar.
Leicester needed to respond at the start of the second half and they did, with Riyad Mahrez's header forcing Everton into a hasty clearance.
Ulloa should then have done better with a free header from Paul Konchesky's cross, but the clear-cut chance they were looking for came in the 65th minute.
Unfortunately for Leicester, it fell for Schlupp, who had just come on as a substitute. Dean Hammond's through-ball was perfectly weighted, but Schlupp got his shot all wrong and it flew a long way over the bar.
Mahrez was the next to try his luck, winning a free-kick 25 yards out for a trip by James McCarthy and then seeing it deflected just wide.
Ulloa's debut ended when he was brought off to loud cheers in the 78th minute and replaced by Wood.
Barry, who had picked up a soft booking for a foul on Ulloa, was a little fortunate to stay on the pitch when he pulled back the lively Mahrez.
And it was Mahrez who set up the equaliser, jinking his way to the edge of the box before firing in a shot that hit Phil Jagielka and fell to Wood.
It was no more than Leicester deserved for their attacking intent and, although Everton tried to respond, it was too late.