With Roma defeating Qarabag both teams needed to prevail in the encounter in West London but, after an own goal from Stefan Savic cancelled out Saul Niguez's opener, neither side could find a winner during the closing stages.
While Atletico will drop into the Europa League, Chelsea face the prospect of playing either Barcelona or Paris Saint-Germain at the last-16 stage of this competition after Christmas.
Atletico began brightly and after a sustained spell of possession at the start of the game, former Chelsea forward Fernando Torres was able to get a shot away inside the area, only for it to be deflected just over the crossbar by a defender.
Chelsea needed 17 minutes to get their first effort on goal and it proved to be their best strike of the opening 45 minutes with Alvaro Morata curling marginally wide of the post from 20 yards out.
A minute later, Morata fired a shot from an acute angle which Oblak clawed away from goal, before Davide Zappacosta was the next to try his luck with a low effort which was parried to the right of the post by the Atletico stopper.
The visitors ended the first half on top but they were not doing enough with the ball in the final third, and Diego Simeone needed to order a change of approach after the restart.
Antoine Griezmann struck a free kick too close to Thibaut Courtois during the first moments of the second half, but Chelsea should have taken the lead through Andreas Christensen, who sent a free header too close to Oblak.
Eden Hazard soon fired wide before Oblak prevented a Morata header from finding the top corner, but Atletico wasted no time before going back up the other end and striking the post through a 25-yard effort from Filipe Luis.
From their next attack, the visitors took the lead thanks to Saul, who was perfectly placed to send a header into the far corner after a smart flick from Torres at the near post.
Antonio Conte wasted little time before throwing on Pedro, and the substitute was instrumental in leading Chelsea's response before Christensen sent a header wide of the post after getting on the end of a driven cross.
Pedro then came within inches of scoring a marvellous equaliser, with the Spaniard's flick from a Moses cutback being cleared off the line by Jose Gimenez, who collided with Oblak for his troubles.
However, Chelsea's pressure finally paid off with 15 minutes left as Hazard danced past a defender before watching his low cross get deflected into the roof of the net by the unfortunate Savic.
Less than 60 seconds after equalising Chelsea should have gone ahead when Cesc Fabregas played in Morata but, despite having just Oblak to beat, the striker drove the ball straight at the goalkeeper.
The Premier League outfit continued to apply pressure and Willian should have scored for the second European game running when being teed up by Hazard, but his first-time effort from 12 yards went high into the stands.
Michy Batshuayi almost won it for Chelsea in added-on time but, after his 20-yard drive went just wide of the far post, the Blues had to be content with a draw which could lead to a meeting with one of European football's big guns after Christmas.
CHELSEA (3-4-2-1): Courtois; Azpilicueta, Christensen, Cahill; Moses, Kante, Bakayoko (Pedro 64'), Zappacosta (Willian 73'); Fabregas, Hazard; Morata (Batshuayi 81')
ATLETICO MADRID (4-5-1): Oblak; Gimenez (Vietto 79'), Savic, Lucas, Filipe Luis; Thomas, Gabi (Correa 79'), Koke, Saul, Griezmann; Torres (Carrasco 57')