Garde was confirmed as Villa's new manager earlier on Monday following the departure of Tim Sherwood, who parted company with the club eight days ago after taking just four points from the opening 10 league games of the season.
Former Lyon coach Garde was in the stands at White Hart Lane, but saw Mousa Dembele give Tottenham the lead after just three minutes with his second goal in as many games after scoring in the 5-1 win at AFC Bournemouth last time out.
Dele Alli's low finish just before half-time looked to have wrapped things up for Tottenham but a late show of spirit enabled Villa to get back into the game.
Jordan Ayew's deflected 79th-minute strike halved the deficit and gave the Midlands club hope of snatching a point, but that was snuffed out in stoppage time as Harry Kane wrapped up the victory for the home side.
Tottenham held firm to go fifth in the table, while Villa remain rooted to the bottom ahead of Garde's first match in charge at home against leaders Manchester City on Sunday.
Dembele immediately provided Garde with an indication of Villa's problems as he outmuscled Ciaran Clark after collecting the ball on the left-hand side and fired a low shot under the legs of Brad Guzan at the near post.
Villa went close to responding when Scott Sinclair forced Hugo Lloris into action with a curling effort before Kane - fresh from scoring a hat-trick in the rout at Bournemouth - saw a deflected strike tipped over by Guzan at the other end.
Danny Rose went close from the subsequent corner and Villa's cause was dealt a further blow when Ashley Westwood was forced off through injury after being caught by the elbow of Dembele.
And the visitors were given a mountain to climb in first-half injury time, Alli firing into the bottom-left corner after Joleon Lescott headed a left-wing cross from Rose straight into his path.
Tottenham appeared to be cruising to victory in a largely low-key second half but, after Leandro Bacuna had struck the post, Ayew set up a nervy finish when he pounced on some poor defending from the hosts with a strike that deflected in off Jan Vertonghen.
Fellow substitute Rudy Gestede headed wide before Ayew jangled the home fans' nerves further by bending an effort narrowly over.
However, Villa could not take advantage of their superiority in the final minutes and were punished for not taking their chances in stoppage time when Kane found the top-right corner as Tottenham moved to within five points of City.
