Heurelho Gomes looked to have gifted the champions the unlikeliest of victories when he clattered into Ramires having already handed Drogba an equaliser with a howler, but he redeemed himself by saving the substitute's spot-kick.
It meant Chelsea failed to regain top spot and stayed fourth, level on points with Manchester United, who have two games in hand.
The manner of the Blues' comeback may ease the pressure on manager Carlo Ancelotti, who continues to insist he has the full backing of billionaire owner Roman Abramovich.
However, Spurs had led through Roman Pavlyuchenko's ninth goal of the season and could easily have won but for their erratic goalkeeper.
Ancelotti got the assertive start he would have wanted, as the recalled Nicolas Anelka dragged wide, Michael Essien twice shot too close to Gomes from distance and Branislav Ivanovic blazed a 30-yard free-kick over the top.
Chelsea also restricted Spurs to one Gareth Bale shot off target until all their good work was undone by some truly awful defending in the 15th minute.
Benoit Assou-Ekotto played a ball to Jermain Defoe in the left channel and the striker was allowed to turn and square for Pavlyuchenko, who John Terry allowed to collect the pass and rifle left-footed beyond Petr Cech.
Salomon Kalou twice went close to levelling, first stretching to send Ramires' cross over the top and then heading straight at Gomes from six yards.
Neither side was holding back in the tackle and Paulo Ferreira was adjudged to have gone too far on Bale just past the half-hour.
Pavlyuchenko maybe should have punished Ferreira when he volleyed Bale's resulting free-kick wide from eight yards.
And Assou-Ekotto was also nearly made to pay for a foul on Ferreira that got him booked when Kalou glanced Florent Malouda's free-kick just past the post 10 minutes before the break.
The same pair combined moments later, with an unmarked Kalou unable to keep his header down from his team-mate's cross.
Anelka did have the ball in the net two minutes from the break but he was rightly ruled offside.
Ancelotti threw on Drogba for John Obi Mikel during the interval but it was Spurs who twice went close to doubling their advantage when Defoe and Luka Modric both shot narrowly wide from the edge of the area.
Drogba did soon test Gomes with a low drive the latter tipped wide moments before the goalkeeper reacted brilliantly to prevent Wilson Palacios' header looping into his own net.
Tottenham manager Harry Redknapp responded to the spell of Chelsea pressure by bringing on Peter Crouch for Defoe.
Palacios looked to have put Pavluchenko through in the 64th minute but the Russian's touch let him down, while Ramires did well to prevent Crouch nodding in Alan Hutton's cross.
Ancelotti sent on Daniel Sturridge for Kalou for the final quarter as Chelsea went for broke and it paid off, but only due to a horrible error from Gomes.
Drogba held off Dawson before lashing a volley the keeper got both hands to but parried into his own net.
Essien saw yellow for a foul on Modric and Sturridge went close to hooking the visitors ahead from a corner.
Frank Lampard - a surprise substitute after three and a half months out injured - then entered the fray, replacing Malouda for the final 13 minutes.
Spurs followed with Robbie Keane for Pavlyuchenko moments before Aaron Lennon flashed wide after more poor Chelsea defending.
Palacios was hurt late on following a challenge from Ashley Cole and was replaced by Sandro.
And just as a draw looked certain, Gomes made another huge error of judgment by clattering into Ramires to concede a 91st-minute penalty.
Drogba stepped up but shot far too close to Gomes, who redeemed himself.
There was still time for Drogba to be booked for scything down Hutton before the final whistle went.