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Frank Lampard's side had experienced a run of five defeats in six in all competitions, and was soon behind when Rashford excellently set up Antony to put United 1-0 up.
Although Everton levelled when Conor Coady took advantage of a David de Gea howler to stab in, more silky work from Rashford restored United's lead as Coady turned his cross past Jordan Pickford for an own goal.
In-form United needed a reprieve in the shape of the video assistant referee (VAR) to deny Dominic Calvert-Lewin a second equaliser, but Rashford grabbed a goal for himself with a stoppage-time penalty as the Red Devils deservedly advanced to maintain interest in silverware on four fronts.
United was ahead after just four minutes, with Rashford driving past his man before finding a pinpoint cross for Antony to poke home, and it could have been 2-0 moments later when Anthony Martial sent a dipping effort just past the post.
But Everton nearly levelled when Demarai Gray's long-range drive crashed off the woodwork, and the Toffees would soon find an equaliser as De Gea somehow let the ball through his legs from Neal Maupay's cross before Coady tapped in.
Rashford played a big role again for United's second, however, brilliantly powering to the byline before his low delivery was turned into his own net by Coady.
The introduction of Everton striker Calvert-Lewin almost reaped dividends when he got on the end of Gray's cross to seemingly level, only for the VAR to spot the winger's offside in the build-up.
There was time then for Ben Godfrey to tangle with Alejandro Garnacho and give Rashford an opportunity from 12 yards, which he confidently took to cap a brilliant display and make sure of United's place in the draw.
