The result eases pressure on Spurs manager Nuno Espirito Santo after the team suffered three consecutive defeats in the league.
Spurs move up to eighth position in the table while Villa drop to 10th place.
"We needed a win but a well-deserved win. We did a good match against tough opponents. Important for us, the boys and the fans. We reacted well after conceding and positive in a lot of ways," Nuno said.
"Many times we conceded and go flat but this time the team did well. We missed chances to kill the game."
The hosts did take the lead four minutes later when Son Heung-Min set up Hojbjerg to score with a composed finish.
Villa made Spurs rue those missed opportunities and equalised after 68 minutes when Watkins converted Targett's cross.
However, Spurs went back in front three minutes later Son provided another assist as Targett bundled the ball into his own net.
Kane, who scored a hat trick against NS Mura in the Europa Conference League in midweek, had chances to kill the game off but was unusually hesitant and the England striker's six-game streak without a Premier League goal is his worst run since the 2015-16 campaign.
"I didn't think there was an awful lot in the match," Villa boss Dean Smith said. "They had the outstanding player on the pitch in Son Heung-Min, he set up the winner and caused us problems all game."