Chelsea opened the scoring on the half-hour mark when Alvaro Morata headered into an empty net after Hugo Lloris mistimed his jump.
However, Christian Eriksen levelled on the verge of half-time and a second-half masterclass saw Spurs take all three points back to home.
The victory puts Tottenham eight points clear of the Blues, just two behind Liverpool and with a game in hand of the Reds.
Dele proving his class
Many have doubted Alli this season, having not quite lived up to the high standards he had set over the past two seasons.
Only playing a matter of minutes during England's two friendly games, the 21-year-old looked out of favour to Jesse Lingard and Raheem Sterling.
Despite the criticism, Alli silenced his doubters with a match-winning brace at the bridge.
Eric Dier's brilliantly lofted ball was cushioned by Alli before lobbing the ball over stand-in goalkeeper Willy Cabellero to give Spurs the lead.
Only four minutes, Alli provided Spurs with a two-goal cushion after netting from a scramble in the Chelsea box.
A fitting celebration followed from Alli as he ran down the touchline with his hand cupping his ear - silencing those who doubted him.
For such a young player who has received a great deal of criticism this season, some fair and some unjustified the 21-year-old put in a much improved, mature performance.
Tactical masterclass from Poch
A half-time change in tactics from Mauricio Pochettino proved the turning point in which Spurs won the game.
The Argentine set Spurs up in a 4-2-3-1 formation with Erik Lamela, Alli and Eriksen behind Heung-min Son.
Eriksen began the game on the wide left of the trio but was moved into the centre in the second-half in order for the Dane to have a greater influence on the game.
Heung-min Son was fairly ineffective up top by himself and was therefore moved out to the right wing to provide Spurs with a greater attacking threat due to the South Korean's pace.
This change paid off for Spurs' third goal as Eriksen's through ball found Son who beat Marcos Alonso for pace which Alli then slotted home.
The change made by Pochettino also stopped Chelsea exploiting the Tottenham backline, Alonso and Victor Moses proved a constant threat throughout the first-half - with Moses grabbing an assist.
Yet in the second-half, the Chelsea full-backs had virtually no influence on the game as Tottenham's wide men provided greater cover for Kieran Trippier and Ben Davies.
For nearly three decades, Spurs had been unable to win a league game at Stamford Bridge.
Despite leading in the previous two meetings before Sunday, the Lilywhite's threw away their lead on both occasions.
The 3-1 victory over the Blues means a great deal more than just the three points for Spurs, it is another mental barrier overcome for the Pochettino project at Spurs.
For years, Spurs have struggled away to their 'top-six' rivals and although one win does not dramatically change their poor record it is certainly a step in the right direction.
One by one, Spurs are ticking off milestones under Pochettino's long-term plan and surely silverware is the next to be achieved.