Atletico scored their first Champions goal after 66 minutes through a sublime volley from Argentine striker Sergio Aguero to go in front.
Ivorian marksman Drogba, back in action after a three-match European suspension, restored parity on 82 minutes after a superb solo effort and added the second six minutes later as Chelsea appeared close to snatching a valuable win at Vicente Calderon.
A frenetic end saw Aguero notch his second of the night two minutes from time with a curling free kick to secure a share of spoils.
Chelsea are on top of Group D with 10 points, one ahead of second-placed Porto who also booked a last-16 berth after defeating APOEL Nicosia, who lie at the bottom with just one point, 1-0.
Colombian striker Falcao scored what proved to be the winner with six minutes remaining.
The struggling Atletico, who are third with two points, will be looking to hold onto their position to claim a Europa League spot.