After going down to 10 men when Yves Bissouma was sent off just before half-time, Spurs' winner came from Micky van de Ven, who poked home his first goal for the club on 52 minutes after James Maddison cut the ball back from a short corner.
Dutch defender Van de Ven said it was an "amazing feeling" to score the match-winning goal.
"On the way to the stadium on the bus Sonny [Son Heung-min] told me: 'You're going to score your first goal today'," he said on TNT Sports.
"I didn't think so. I said: 'Maybe, who knows.' But it is perfect because now I have scored my first goal.
"I think we showed a great mentality as a team."
Luton had captain Tom Lockyer's low header chalked off when Elijah Adebayo was deemed to have fouled Cristian Romero as he headed Mads Andersen's free-kick across goal.
A more positive approach in the second half led to them creating plenty of opportunities but Tottenham held on to leave Rob Edwards' side without a win at home this season.
Ange Postecoglou's Spurs are now unbeaten in eight Premier League games and sit two points ahead of Manchester City, who face Arsenal on Sunday.
Remarkably, the home side made it through the opening 20 minutes without conceding after Spurs had peppered the Luton goal.
Richarlison twice had opportunities in the opening five minutes, but poked his first effort from Maddison's whipped cross over the bar before having a low drive saved by the boot of Thomas Kaminski in the Luton goal.
Pedro Porro then struck a shot wide and a poor Tom Lockyer clearance was snatched at by Son.
Bissouma had already ensured he would miss Spurs' next game by picking up a fifth booking of the season for dragging down Chiedozie Ogbene.
But the midfielder, who has had an impressive start to the campaign, then picked up a second yellow for diving on the edge of the Luton area.
After taking the lead in the second half, Postecoglou's side barely created any further chances, and took off the attacking threat of Son and Maddison in favour of the more defensive options Oliver Skipp and Emerson Royal.
But under the Australian manager Spurs have shown resolve and belief, and more of that was on display on Saturday as they got the job done against 17th-placed Luton.
Postecoglou praised his side's "character and resilience" but said they should have been "three or four up" after the first half.
"We started really well but then you get the red card and it is a different type of game," he told TNT Sports.
"We are really tight as a group, which you see because, not only are we winning games, but apart from the football which has been good, the resilience we have shown has been great.
"The players want to change the destiny of this football club and that is what they are trying to do on the pitch."
Should the match between Manchester City and Arsenal result in a draw on Sunday, Spurs will go into the international break top of the league.
Edwards' Luton have managed just four points in eight games so far this season.
After a dominant opening period from the visitors, which saw Luton produce just 11 passes, the home side began to make the one-sided affair more of a contest but could not manage a shot on target until their 40th-minute disallowed goal.
With a one-man advantage after the break they missed a glorious opportunity when Ogbene's measured pass seemed to have teed up Adebayo, who only had to slide in and connect for a tap-in, but could not sort his feet out and muddled it past the post.
Alfie Doughty also scuffed a good opportunity wide and Carlton Morris was denied by the arm of Spurs keeper Guglielmo Vicario as Luton probed for an equaliser.
The Hatters had scored a late goal before eventually losing to Burnley last time out but could not make any of their chances count and looked sapped of energy by the final stages.
-- Courtesy of BBC Sport