Arsenal’s bright start was rewarded on the half-hour when Saka fired in from the edge of the area after Villa had failed to clear a set-piece.
It made Arsenal the third team to reach the 2,000-goal landmark after Manchester United and Liverpool.
Villa pressed for an equaliser after the break and the closest they came was on 68 minutes, when Ollie Watkins’ shot deflected off Kieran Tierney and hit the near post.
A fifth consecutive away win for the first time since 2015 takes fourth-placed Arsenal four points clear of Man Utd, with 54 points.
Villa are ninth with 36 points.