Just four days after suffering their first Premier League defeat of the season at the hands of the same opposition - when the Reds edged a five-goal thriller - the Blues enjoyed a measure of revenge following another goalfest in west London.
However, the Blues had to come from behind three times and needed a last-gasp Eden Hazard penalty to take the match into overtime.
United were gifted an opening goal on 21 minutes when Oriol Romeu was robbed by Anderson who set up Ryan Giggs to fire a low curling left foot strike beyond the reach of keeper Petr Cech.
Yet the Blues were level from the penalty spot eight minutes later: Alexander Buttner tripped Victor Moses and David Luiz smashed the ball into the bottom left-hand corner of the net although keeper Anders Lindegaard managed to get a hand on it.
Javier Hernandez was at his predatory best to fire United back in front two minutes before half-time.
After Anderson pounced on a Luiz mistake, the Brazilian spotted Hernandez who raced clear of the home defence before slipping the ball past an advancing Cech.
Chelsea were level for a second time seven minutes after the break when Juan Mata's corner was crisply headed home by Gary Cahill.
However, Nani superbly chipped United back in front just before the hour mark after playing a quick one-two with Anderson - who collected his third assist of the night.
And with 93 minutes on the clock, a clumsy Scott Wootton bundled over Ramires in the box and Hazard cooly slotted home Chelsea's second penalty of the evening to take the tie into extra-time.
Wooton was also at fault when Chelsea went ahead for the first time just seven minutes into extra-time.
The youngster's weak mis-directed header back to his keeper was gratefully gobbled up by Daniel Sturridge who made amends for his previous misses to fire Chelsea 4-3 ahead.
And four minutes from the end of extra time, Hazard set up Ramires who rounded Lindegaard to slip the ball into an empty net and put Chelsea 5-3 ahead.There was still time for Giggs to score the ninth goal of the evening from the penalty spot, following a foul by Cesar Azpilicueta on Hernandez in the 119th minute, yet it came too late to save the visitors as the Blues made it through to the quarter-finals of a competition they used to call the League Cup.
|Chelsea||Team Statistics||Manchester United|
|1||1st Half Goals||2|
|10||Shots on Target||9|
|15||Shots off Target||5|
|58.6||1st Half Poss.||41.4|
|56.2||2nd Half Poss.||43.8|