It was a frustrating night for Fulham, who were forced to make a late change before kick-off when Willian was injured in the warm-up, and then Harry Wilson, who had scored in both their past two wins, limped off injured on 17 minutes.
Villa, who had scored in all 19 of their previous Premier League contests since Emery took charge, extended that run to a competition-best 20 matches with Tyrone Mings' flicked header after 21 minutes. It was the defender's first Premier League goal since November 2021.
Villa extend their unbeaten run to 10 matches, winning eight, and have won five straight home PL matches without conceding for the first time.
It takes them on to 54 points, one more than Tottenham Hotspur, while Fulham stay ninth on 45 points.
Vardy denies Leeds big win
Despite trailing, Leicester enjoyed more of the possession, but Leeds continued to frustrate the Foxes, efficiently and consistently breaking up play.
A tactical tweak at half-time saw Harvey Barnes pushed further forward and he curled inches wide of the upright as Leicester pressed for an equaliser.
But it was Vardy, who replaced Barnes with 20 minutes left, who came to the rescue with an 80th-minute leveller, a cool finish after being set up by James Maddison, back in the team following illness.
The drama was not over, however, as Daniel Iversen made two superb saves before Patrick Bamford missed from four yards in a pulsating finish.
Neves stars as Wolves end Hodgson's run
Palace, unbeaten in four matches since Roy Hodgson returned to Selhurst Park, were denied an equaliser just before half-time when Jose Sa reacted quickly to keep out Albert Sambi Lokonga's snapshot from close range.
Wolves move up to 13th spot, level on 37 points with Palace, who are one place above on goal difference.
Mouthwatering action continues on Wednesday
Elsewhere, a rejuvenated West Ham United face Liverpool at London Stadium, Nottingham Forest aim to ease their relegation fears at home to Brighton & Hove Albion, while local pride is at stake in the west London derby between Chelsea and Brentford.