Garde, named as Villa's new boss earlier in the day, watched from the stands as they slumped to a seventh straight league defeat to leave them with only four points from 11 matches.
Tottenham extended their unbeaten league run to 10 to move up to fifth in the table, above West Ham United on goal difference, one point off the top four.
The hosts took the lead when Mousa Dembele forced his way into the penalty area and scored with an angled shot after three minutes and youngster Dele Alli powered in a low shot from the edge of the penalty area on the stroke of halftime.
Jordan Ayew pulled one back for Villa after 78 minutes to give them a glimmer of hope before Harry Kane sealed the victory for Spurs with a curling shot to round off a superb move in the third minute of stoppage time.
Garde, sitting in the front row of the director's box at White Hart Lane, would have taken heart from a spirited late push from Villa but they will need to show a far greater improvement if they are to avoid relegation.
Villa, with caretaker manager Kevin MacDonald in the dugout, lacked confidence and rarely troubled Spurs, although Leandro Bacuna struck the post with a shot just before Ayew scored.
"For the new manager, the shift the players put in shows that there is something for the manager to work with," MacDonald said.
Garde's first match in charge is against leaders Manchester City on Sunday and his first six matches are all against teams in the top half of the table.
With Tottenham unbeaten in the league since losing to Manchester United on the opening day of the season, hopes of a challenge for a top-four finish are rising, but manager Mauricio Pochettino is not getting carried away.
"It's not important. The next game is important and we need to keep our focus and win the next game. Our start is good but that's nothing," he said.
Tottenham's next game, away to second-placed London rivals Arsenal on Sunday, will reveal more about their credentials.
(Reporting by Mike Collett; editing by Martyn Herman)
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