Victory at Old Trafford also lifted United to within four points of fourth-place Manchester City, on whom they have a game in hand, ahead of the derby at the Etihad Stadium on 27 April.
It was a personal triumph for United manager Jose Mourinho, who saw his side thrashed 4-0 at Chelsea in October and was branded "Judas" by fans of his old club during last month's FA Cup defeat at Stamford Bridge.
Chelsea have now lost twice in four games, having gone down 2-1 at home to Crystal Palace earlier this month, which will give in-form Spurs fresh hope that the Blues may yet collapse as the finishing line nears.
The sides meet at Wembley in the FA Cup semi-finals next weekend in what now looks a significant test of Chelsea's mettle. United, meanwhile, will approach Thursday's Europa League quarter-final second leg at home to Anderlecht, which is poised at 1-1, on the back of their best display under Mourinho.
Chelsea were without goalkeeper Thibaut Courtois due to an ankle problem, while Marcos Alonso succumbed to a virus, bringing Asmir Begovic and Kurt Zouma in as replacements.
Mourinho set United up to stop Chelsea, with Herrera and Matteo Darmian tasked with the responsibility of tracking dangermen Eden Hazard and Pedro wherever they went.
But despite Herrera's chief role as a stifler, he created United's seventh-minute opener, albeit in contentious circumstances. He seemed to block Nemanja Matic's pass with his right hand, but referee Robert Madley allowed play to continue, enabling Herrera to split the Chelsea defence with a sublime through ball for Rashford.
The 19-year-old raced in behind David Luiz before clipping a shot past Begovic for his ninth goal of the season.
With Chelsea striker Diego Costa spending more time provoking his opponents than taking shots at goal, David de Gea was a virtual spectator between the posts for United.
Costa became embroiled in penalty-box tangles with United centre-backs Eric Bailly and Marcos Rojo and was booked for a late challenge on Paul Pogba.
With the willing Jesse Lingard in support of Rashford, United looked far more lively going forward.
Ashley Young twice spurned presentable shooting opportunities from the edge of the box, while Rashford saw a cross headed onto the roof of the net by Gary Cahill.
Any hopes the visitors had of masterminding a turnaround vanished within four minutes of kick-off in the second half as Herrera doubled United's lead. Zouma's tackle on Young sent the ball rolling into Herrera's path and his drilled effort flicked off Zouma to fly past the grounded Begovic.
Chelsea manager Antonio Conte sent on Cesc Fabregas for Victor Moses, with Pedro moving to wing-back, but still the league leaders toiled, Pedro twice curling shots over De Gea's bar.