Sergio Aguero scored a hat-trick in the last half hour as City escaped what would have been a Cup upset by coming from behind to win 4-2 at home to second-tier Watford.
Watford were still within sight of a money-spinning replay, but with three minutes remaining Kolarov deceived visiting goalkeeper Jonathan Bond from 40 yards to make it 3-2 before Aguero completed his treble at the death.
Beaten Championship play-off finalists last season, Watford went 2-0 up inside 30 minutes through Fernando Forestieri and Troy Deeney to leave free-scoring City at risk of a shock defeat.
Liverpool also avoided a potential upset by winning 2-0 at second-tier Bournemouth as Victor Moses and Daniel Sturridge scored.
Liverpool manager Brendan Rodgers praised his opposite number Eddie Howe for the way that Bournemouth had approached the game.
Premier League Crystal Palace were the biggest name to get knocked out, losing 2-1 at holders Wigan. Ben Watson, who scored the winning goal at Wembley, set Wigan on their way in the 36th minute, with James McClean netting a 78th-minute winner after Aaron Wilbraham drew Palace level.
Sunderland followed up their mid-week success against Manchester United in the League Cup semi-finals by toiling to overcome non-league Kidderminster Harriers 1-0, with Charis Mavrias scoring the only goal in the fifth minute.
Swansea City, Cardiff City, Southampton and Hull City also overcame lower-league opposition, while Everton were due to visit third-tier Stevenage in the late game.