Simon Mignolet blocked Leroy Sane’s cross-shot with his left boot after 19 minutes, David Silva slicing the rebound over the crossbar to spurn City’s first real chance of the game.
Penalty appeals were rejected at both ends, with Aguero challenged by Joel Matip as the Argentine tried to connect with Fernandinho’s low cross, before Sadio Mane collided with Nicolas Otamendi as he shaped to shoot.
Matip headed Milner’s deep free-kick wide after 37 minutes and the game finally exploded into life before the interval.
Milner had to make a brilliant challenge to stop Raheem Sterling from tapping home Silva’s cross, with Fernandinho unable to convert at the back post, before Caballero palmed Roberto Firmino’s curled shot over his crossbar.
Caballero elected to punch the resulting corner but he could only pick out Lallana, whose powerful volley was beaten away by the goalkeeper.
Chances continued to flow in the second half, with Mane’s goal-bound effort deflected over by a Stones block, but Liverpool were awarded a penalty when Clichy clumsily bundled over Firmino.
Former City player Milner tucked away the spot-kick, sending Caballero the wrong way to put the visitors in front, extending City’s run without a clean sheet against Liverpool to 13 games.
Stones had a big chance to rapidly respond for City but he headed De Bruyne’s corner wide. Caballero then did well to save from Firmino after a superb Philippe Coutinho reverse pass sent his fellow Brazilian into the box.
But after 68 minutes City levelled the scores when Aguero tucked away De Bruyne’s cross to continue his record of scoring in all five of his home league games against Liverpool.
Aguero’s one-two with Sane freed the striker for a chance to complete the turnaround but he could not find a finish after losing his balance, De Bruyne slamming the loose ball against the woodwork, then Lallana’s opportunity to claim the points arrived.
Emre Can and Firmino combined beautifully to set Lallana up for a golden chance but the midfielder failed to convert and despite a breathless finale – with Aguero unable to turn in Silva’s centre and blazing over De Bruyne’s chipped cross – the points were shared.