The Azerbaijani minnows were on course for a shock victory at the Wanda Metropolitano, two weeks on from claiming their first ever point in the competition with a goalless draw against Atleti in Baku.
Atletico temporarily recovered as Thomas Partey cancelled out Michel's shock opener 11 minutes into the second half but, even when the opposition were reduced to 10 men shortly after, they could not find a winner.
Victory for Roma over Chelsea elsewhere in Group C leaves the three-time runners-up third in the standings, four points adrift of the Blues and five off the leaders with two games - against the two sides above them - to come.
Atletico made a positive start to the contest as they went in search of a maiden European win at their new 68,000-capacity home, testing Ibrahim Sehic through an Antoine Griezmann strike five minutes in.
The visitors have struggled for goals of late, however, netting four in their last six matches in all competitions, and their profligacy continued as Kevin Gameiro failed to get the better of Sehic when latching on to an Angel Correa back-heeled pass.
Centre-back Diego Godin advanced forward and was the next player to come close, heading over the bar at the end of a Partey cross, while Griezmann - now with one goal in his last eight games for club and country - directed a free kick over the bar from 23 yards out.
Qarabag may have had to soak up the pressure, but they edged the first-half possession count and created the odd opening of their own before taking a shock lead.
The breakthrough moment in the match arrived five minutes before the interval as Spanish midfielder Michel, formerly of Sporting Gijon, got his head to Wilde-Donald Guerrier's set-piece delivery and picked out the top corner.
Atletico's players, who trained at their former Vicente Calderon home on the eve of this match, emerged for the second half well before their opponents.
Things soon started to go in the hosts' favour as, after Gameiro somehow failed to adjust his feet and convert the ball after Sehic parried Correa's shot into his path, Los Rojiblancos had the goal they craved.
Partey curled the ball past the reach of the opposition keeper from all of 25 yards, moments after Stefan Savic produced a superb interception to deny Henrique the simplest of tap-ins.
Qarabag's evening took a turn for the worse three minutes later when Henrique was shown a straight red card for a high-boot challenge on Godin, needlessly catching the centre-back's head to leave his side with 10 men.
Substitute Nicolas Gaitan had a header blocked on the line and Filipe Luis fired wide following some smart interplay, as Atleti - with 30 wins in 36 Euro home games prior to tonight - knocked on the door for a winner.
The closest Atletico came to netting that was through a Gaitan drive late on, helped wide of the target by Sehic, with Savic's second bookable offence in the 88th minute - a needless challenge with his side in the ascendancy - effectively signalling the end of their recovery and a likely group-stage exit.