But Drogba then saw his stoppage-time penalty saved by Heurelho Gomes, who conceded the spot-kick after bringing down Ramires.
Ivory Coast striker Drogba took the penalty even though Frank Lampard, Chelsea's regular spot-kick taker was on the field.
Had the penalty gone in and Chelsea won, the Blues would have gone top.
Instead they stayed fourth, level on points with Manchester United.
But United will go top if they beat leaders Arsenal on Monday.
Tottenham opened the scoring in the 15th minute when the unchallenged Roman Pavlyuchenko struck from inside the box after a pass from Jermain Defoe.
Drogba, on at the start of the second half, equalised 20 minutes from time when his shot somehow found a way through Gomes's hands.
Spurs regained fifth spot from Bolton, who beat Blackburn Rovers 2-1 earlier Sunday.