Chelsea were bidding to become the first team to win 14 successive games within the same English top-flight season, but Alli's goals, both set up by Christian Eriksen, brought them down to earth.
"We were competitive against a team in very good form, one of the best in Europe, and we feel very proud and pleased because the performance was very solid," said Spurs manager Mauricio Pochettino.
Antonio Conte's Chelsea must content themselves with having equalled Arsenal's season record of 13 wins in a row from 2001-02 and they remain five points clear of second-place Liverpool.
But the end of their winning streak will give hope to their pursuers, not least Tottenham, who rose above Manchester City and Arsenal to third place in the table, seven points below the leaders.
"It is a pity to stop this run, but Tottenham is a good team, a really strong team and I think is for sure one of the teams that can fight for the title until the end," Conte told Sky Sports.
"My team are top of the table because we are working a lot. This league is very tough and it is important to continue to work."
Alli, 20, has now scored braces in three successive games and his latest double allowed Pochettino's men to register a fifth win in a row -- their best such run in nearly a year.
It also enabled Spurs to claim a measure of revenge, both for their 2-1 defeat at Chelsea in November and the fiery 2-2 draw at Stamford Bridge last May that ended their title challenge.
Pochettino, meanwhile, could celebrate his 150th Premier League game as a manager with a deeply satisfying victory.
"We are in a good position, third in the table," he said. "But it is a long way to finish the season."
Whereas Spurs had gone with four at the back in November's 2-1 loss at the Bridge, here they effectively matched Chelsea's 3-4-3 shape, which made for an attritional first half at White Hart Lane.
Spurs saw most of the ball, their wing-backs Kyle Walker and Danny Rose carrying the fight, but they struggled to pick holes in Chelsea's rearguard and the visitors posed a threat on the break.
In the fifth minute, Nemanja Matic's lofted pass over Spurs' right-sided centre-back Eric Dier found Eden Hazard in ample space, only for the Belgian to shoot narrowly wide with his left foot.
Victor Wanyama and Eriksen sent low efforts wide from range for Spurs and when the hosts finally got in behind, Dier ghosting in to meet Rose's deep free-kick, Thibaut Courtois was on hand to save.
But in first-half injury time, Spurs struck, Eriksen shaping a cross into the box from the right and Alli, hitherto close to anonymous, leaping to plant a glorious header in the top-right corner.
It was the England midfielder's sixth goal in four games and within nine minutes of kick-off in the second half, he would have a seventh.
Urged on by a typically animated Conte, Chelsea flew out of the traps, Diego Costa testing Hugo Lloris with a skidding strike and Hazard nodding Marcos Alonso's header wide from point-blank range.
The celebrations, though, were Tottenham's thanks to a near carbon copy of their first goal, Eriksen swinging a cross to the back post and Alli outleaping Victor Moses and Cesar Azpilicueta to head home.
Conte made three bold changes, sending on Willian, Cesc Fabregas and Michy Batshuayi for Marcos Alonso, N'Golo Kante and Moses, with Pedro Rodriguez moving to a left wing-back role.
But his players, finally, appeared spent and Eriksen came close to adding a third with a dipping free-kick that narrowly eluded the left-hand upright.