The Championship side defended resolutely but their rearguard was finally breached in the 71st minute when Ben Davies' glancing header from substitute Georges-Kevin Nkoudou's cross was too good for Sam Johnstone in the Villa goal.
Just ten minutes later Spurs confirmed their place in the fourth round when Moussa Sissoko pulled the ball back for Son Heung-min to slide a low shot home.
Spurs enjoyed the majority of the possession throughout the match as Villa sat back but the home team were unable to create many clear-cut openings until the final quarter of the encounter.
In a dreary first half Leandro Bacuna's 25-yard effort was gathered by Michel Vorm while the closest the home side came to scoring was just before the interval when Jordan Amavi almost turned the ball into his own net.
Spurs created the first opening of the second period when Kieran Trippier's excellent cross found Son but he mis-hit his shot and the Villa defence were able to clear their lines before Vincent Janssen headed the former Burnley defender's corner wide.
The Dutch striker was almost immediately replaced by Dele Alli and Spurs looked livelier as a result of the change with Son seeing his 61st-minute shot tipped over the bar for a corner by Johnstone.
The Villa goalkeeper then claimed Harry Winks' deflected shot but Spurs turned on the style in the closing stages to run out comfortable winners.