Daniel Iversen made a stunning reflex save to deny Iwobi early in the first half, and the Danish goalkeeper also kept out efforts including Calvert-Lewin’s shocking miss and Dwight McNeil’s strike from eight yards out.
But he saved his best until late when Everton pushed for the winner and Abdoulaye Doucoure strode forward before unleashing an arrowed low drive which Iversen turned round the post at full stretch.
In between, they face a trip to Wolves – but that too is a tough ask for a side winless in their past 15 on the road and victors in just two of 34 away league games.
The league’s lowest scorers managed to bag two this time, and Dyche will be satisfied to rescue a point, but it could have been so much better with more clinical finishing.
This was a significant chance to get back to winning ways and it started well through Calvert-Lewin’s thumping penalty, but it is difficult to see where the points might come in their remaining games.
They travel to impressive Brighton in their next match, before hosting league leaders Manchester City and rounding off their season at Goodison Park against in-form Bournemouth. Like their opponents, Everton are stuck in a rut. Despite a bright start the Toffees have now won just one of their past 11 games under Dyche, and the eight points gained on that run have not been enough to drag them out of trouble.
They also missed the opportunity to collect back-to-back victories at home for the first time in a year and bear all the hallmarks of a relegation-threatened team.
Despite being on a three-game unbeaten run, the Foxes have won just one of their past 12, picking up only six points in the process.
Leicester have four games remaining to preserve their top-flight status, away at Fulham and Newcastle and home games against Liverpool and West Ham on the last day of the season.
Midfielder Maddison was heavily involved throughout the game as well as the miss from 12 yards, and saw a low drive and curling effort kept out by his England team-mate.
The home side recovered from Calvert-Lewin’s 15th-minute penalty to turn the game around as defender Soyuncu levelled and Vardy rolled back the years with a vintage finish.
But Leicester’s issues lie in defence where a porous backline have now failed to keep a clean sheet in their past 19 games, dating back to November.
An electric atmosphere was generated at King Power Stadium by both the home fans with their ‘clappers’ and the away supporters through their vociferous backing.
Ultimately, Maddison’s penalty miss at the end of the first half turned out to be the crucial moment – the playmaker was made to wait to take the spot-kick before fluffing his effort straight at Pickford.
It would have put Leicester 3-1 up and possibly out of sight, but Sean Dyche’s men clawed out a draw courtesy of Iwobi’s well-taken strike on 54 minutes. The game ebbed and flowed between two sides aiming for top-flight survival but was a missed opportunity in their aim to escape the relegation mire.
It proved costly as the visitors drew level nine minutes after the restart when Alex Iwobi guided in a low finish, but neither side were able to find a winner.
Everton, meanwhile, will also be concerned by a serious-looking injury to captain Seamus Coleman, sustained in the opening period following a collision with Boubakary Soumare. There was still time in a breathless first half for Pickford to save Maddison’s penalty, which was struck straight down the middle by the England midfielder. The Toffees should have levelled before half-time but Calvert-Lewin contrived to miss an open goal from a couple of yards with the ball appearing to strike his heel, before Leicester broke down the other end and Vardy struck the crossbar.
Dominic Calvert-Lewin opened the scoring with just his second goal of the season from the penalty spot, picking himself up to convert after being bundled over by Timothy Castagne.
The lead lasted just seven minutes as the hosts equalised through Caglar Soyuncu, who rolled home a cool finish from Wout Faes’ knockdown.
And Leicester turned the game around after 33 minutes as the sprightly Jamie Vardy latched on to James Maddison’s through ball before rounding Jordan Pickford and stroking home. The two clubs started the game in the bottom three but a point was enough to drag Leicester out of the drop zone on goal difference, while Everton stay 19th, one point behind 17th-placed Leeds.