The damage was done in a clinical 14-minute spell at the end of the first half as Pogba set up Herrera for the opener and then added a second himself with a header of his own.
Chelsea had only failed to score in one of their last 50 cup ties on home soil heading into this tie - a repeat of last season's final - but they did not muster a shot on target between the 11th minute and full time.
Maurizio Sarri's subs were also jeered as frustrated supporters saw their side offer little in the way of a fightback, with the pressure on the Italian continuing to build ahead of the EFL Cup final clash against Manchester City next weekend.
Both keepers were heavily involved in a lively opening quarter to the match, as Sergio Romero produced a double save to deny David Luiz and Pedro after Chris Smalling tested Kepa Arrizabalaga at the other end with a back-post header.
Herrera was next to come close, stinging the gloves of Kepa with his 30-yard drive, and it was the Spaniard who would open the scoring for the visitors just after the half-hour mark.
Chelsea's backline were caught out by Pogba's teasing delivery from deep, which Herrera watched all the way to guide past Kepa after creeping in around the back.
The home side, victorious in each of their last four FA Cup clashes with United prior to today, saw a couple of openings come and go for Gonzalo Higuain as they attempted to respond to the setback.
A second goal was added to United's tally before the break, however, courtesy of a quick counter that again left Chelsea exposed.
Pogba was one of three players waiting for the ball in the middle and he made no mistake when Marcus Rashford picked him out, powering a header into the back of the net from 10 yards.
United felt that they should have had a penalty early in the second half as Luiz's vital interception looped into the air and hit Cesar Azpilicueta on the shoulder, though Chelsea were also hard done by minutes later as Nemanja Matic avoided a second yellow following a succession of fouls.
Eden Hazard looked like being Chelsea's best route back into the match, cutting in from the right and coming close through a deflected shot, but the hosts otherwise lacked ideas in the final third.
There was to be no more shots of note in the remaining 25 minutes of the match as United saw things through in a professional manner to gain revenge for last season's defeat at Wembley.
CHELSEA (4-3-3): Kepa; Azpilicueta (Zappacosta 82'), Luiz, Rudiger, Alonso; Kante, Jorginho, Kovacic (Barkley 71'); Pedro (Willian 58'), Higuain, Hazard
MANCHESTER UNITED (4-3-1-2): Romero; Young, Lindelof, Smalling, Shaw; Herrera, Matic (Pereira 76'), Pogba; Mata; Lukaku (Sanchez 73'), Rashford (McTominay 93')