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It was a measure of revenge for United, resurgent under interim coach Ole Gunnar Solskjaer, who lost to Chelsea in last May's Wembley final under their former boss Jose Mourinho.
Chelsea's fragile defence was split with crosses from either flank. Paul Pogba supplied the first in the 31st minute after a fast move down the left and Herrera ghosted into the box to nod the ball down past stranded goalkeeper Kepa Arrizabalaga.
Until that moment a lively game had been evenly poised, with a host of chances at either end. Chelsea's hopes were crushed just before halftime, however, in a speedy United counter down the right, with Rashford crossing for Pogba to head powerfully home between defenders Cesar Azpilicueta and Antonio Rudiger.
The fifth-round defeat piled more pressure on Chelsea coach Maurizio Sarri, whose side were hammered 6-0 last week in the Premier League by United's neighbours Manchester City.
Chants of "sacked in the morning" broke out in the second half as the hosts failed to break United down. There were boos from some home supporters at the final whistle while Solskjaer celebrated with his players in front of the travelling fans.