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Everton had looked to have earned a draw when defender Joleon Lescott scored his team's second equalizer just 30 seconds earlier before Young's decisive goal, but Young burst through the middle, slalomed past Lescott and hit a right-foot shot past goalkeeper Tim Howard.
The victory lifted Villa to 28 points, just one fewer than Arsenal, which occupies the last of the four lucrative Champions League qualifying spots.
Both Young's goals came from mistakes by the home side, the first a misplaced pass by Phil Jagielka and the second a sloppy touch by midfielder Mikel Arteta. Villa's other goal came from Steve Sidwell, who scored after just 34 seconds for the quickest goal so far this season.
Everton's second loss in eight matches leaves it in eighth place on 22 points, one place and one point behind Portsmouth, which drew 1-1 at bottom side West Bromwich Albion.
West Brom captain Jonathan Greening put the Baggies ahead in the 39th from a rebound after Chris Brunt shot against the bar, but Greening reacted first to the rebound and tucked away his first goal of the season. But England striker Peter Crouch equalized 19 minutes later with a deflected shot from just outside the area.
West Brom stayed last with 12 points and could slip four points from safety if Tottenham wins at West Ham on Monday.