But interim boss Emerse Fae now has a chance to guide the Elephants to a third continental title, adding to triumphs in 1992 and 2015, when they face three-time winners Nigeria in Abidjan on Sunday (20:00 GMT).
An injury to Neto curtailed Wolves' momentum but Fabian Schar headed the last chance wide and both teams had to accept a point. The result means Eddie Howe's team miss the chance to move above Aston Villa and remain sixth.
The visitors had two more efforts ruled out for offside before Marcus Alonso won it for them in injury time, meeting Timo Werner's pull-back and sending the ball looping over Ederson.
It was 2-2 going into the second leg and Kevin de Bruyne curled in a low shot to put City ahead on aggregate. Striker Sergio Aguero had missed the chance to put the hosts ahead on the night when he sent a penalty wide.
Boro's best chance came when Alvaro Negredo's 78th-minute volley was saved. It was the only time the home side forced Blues keeper Thibaut Courtois into a serious stop as they lost for the first time in four matches.
Virgil van Dijk's header in the first half and Luis Diaz's header after the interval were as close as Liverpool came to the breakthrough. Even Diogo Dalot's red card in stoppage time did not help them.
But City's Argentine striker Sergio Aguero missed a chance to equal the club's all-time scoring record when his penalty midway through the second half was saved.
A thrilling finale against West Ham last time out in the league was quickly forgotten three days later when the Hammers went to Old Trafford and won, and Ole Gunnar Solskjaer faces further scrutiny after Kourtney Hause's header secured a 1-0
Marco Silva's side were in total control by half-time after Willian added a second from Andreas Pereira's cut-back and Tosin Adarabioyo out-jumped Kurt Zouma at a corner to head in the hosts' third.
Fernandinho clipped the post in a rare City chance, but Real dominated as Bale hit the bar and Joe Hart saved from Luka Modric and Cristiano Ronaldo. Real saw out the win to meet Atletico Madrid in the final later this month.
Pedro added a third goal deep in injury time when his shot hit Steve Cook and the ball span over the line. Bournemouth's best chance saw substitute Benik Afobe denied by Thibaut Courtois but the Cherries ended up well beaten.
Even the second-half introduction of Alexis Sanchez - featuring for the first time since suffering a hamstring injury in November - could not spark Arsenal, who lost at home in the league for the first time since the opening weekend of the season