United boss Erik ten Hag made a statement with a strong starting line-up for the third-round tie, able to call upon Antony and Anthony Martial up front while Lampard had to leave Dominic Calvert-Lewin on the bench due to fatigue.
Despite lining up in a 5-4-1 formation, Everton looked porous at the back and were under the cosh instantly.
It took just three minutes for Rashford to burst down the line, shown onto his left by Ben Godfrey, and roll a ball deep into the six-yard box for Antony to slide home.
Martial threatened a second as Everton looked at sixes and sevens, and yet that was nothing compared to the hapless defending that brought about their equaliser.
Demarai Gray’s hit from distance smacked the post and off the back of David De Gea, who fortuitously saw the ball land the right side of his woodwork and out for a corner. Yet the goalkeeper found no favours in his dodgy footwork from Neal Maupay’s cross into the near post moments later, which squeezed between his legs for Conor Coady to nudge home.
The quality of defending failed to increase as Jordan Pickford smartly denied the all-too-free Martial an instant reply.
While Everton and Demarai Gray in particular threatened on the break, United were generally on top with plenty of threat in the transition and when building up more gradually.
The visitors crowding the box often meant the final ball was deprived of finding its target, so it was little surprise when Christian Eriksen fired off from distance and nearly beat Pickford with a fine, but just-too-high, curling effort.
Everton suffered a nightmare start to the second half as Alex Iwobi suffered a painful ankle injury when on the end of a tough-but-legal challenge from Tyrell Malacia. Evidently in agony, the midfielder held his shirt over his face as he was taken off on a stretcher.
The Toffees had simply not got going again when Rashford darted down the left, dribbled along the byline and put in a low cross that Coady deflected into his own net.
United enjoyed a long spell on top but failed to really test Pickford in search of a third to kill the tie and were almost punished just after the hour mark when Seamus Coleman’s cross was toed by De Gea into Vitaliy Mykolenko’s path, only for his shot to be blocked off the line.
Their biggest escape came shortly after Calvert-Lewin was introduced for the final 20 minutes, the striker smartly bundling home Gray’s cross after the winger left Malacia for dead. However, the 9,500 away fans’ cheers were cut short by a marginal VAR call in the build-up.
United dropped to a back five late on as Lampard threw on attackers but a legal equaliser was not forthcoming with Ben Godfrey’s challenge on Alejandro Garnacho handing Rashford the chance to seal it at the death.
The striker duly rolled his spot-kick home to seal his side’s seventh win in a row and their spot in Sunday’s draw while Lampard’s Everton await a crunch relegation six-pointer against Everton a week on Saturday.