The Reds hit the woodwork three times and had an effort cleared off the line before and after substitute Joel Matip eventually headed in late on to make it 1-1 at full-time - cancelling out Raheem Sterling's early opener.
In the subsequent shootout, City goalkeeper Claudio Bravo saved Georginio Wijnaldum's spot-kick and that proved the difference as Liverpool were beaten 5-4 on penalties.
See a selection of the best images from the match in the gallery below...