Despite claiming one point from their opening two games against AS Roma and Chelsea, Diego Simeone's Atletico side – finalists in two of the last four seasons – were expected to make light work of unfancied Qarabag.
Yet the Spanish side lacked a clinical edge, with Yannick Carrasco and Antoine Griezmann missing good chances in the first half before the Frenchman had an effort disallowed for offside after the break.
Azeri side Qarabag looked dangerous on the counter-attack but their late push for an unlikely winner was all but ended when Dino Ndlovu was dismissed having received a second yellow card for a dive in the 75th minute.
Chelsea lead Group C with six points and host second-place AS Roma, on four points, later on Wednesday. Atletico are two points adrift in third and can ill-afford to lose any more ground on the top two sides.