Solskjaer’s position has looked increasingly precarious following recent Premier League defeats to Liverpool and Burnley, and a third setback in the space of eight days to a side sitting third bottom in League One would have dramatically escalated the situation.
But Harry Maguire’s screamer was followed by cool finishes from Diogo Dalot and Jesse Lingard as United scored three times in six first-half minutes while Phil Jones and Anthony Martial also found the net before the interval.
Matters hardly improved for the hosts after the break as Mason Greenwood struck from the spot, meaning the sides’ second competitive meeting proved even more one-sided than their first – when United won 5-0 in 1977.
Tranmere had reached this stage by beating a callow Watford three days earlier and, while Solskjaer made six changes to the team beaten by Burnley in midweek, United’s XI was not light on experience as Victor Lindelof, Luke Shaw and Lingard were among those to come back into the side.
Much of the build-up had centred on the boggy nature of the pitch, which had several bare patches, while both flanks were heavily sanded, but any United concerns soon evaporated once they had taken the lead.
In a raucous atmosphere and with Morgan Ferrier making a nuisance of himself early on, with Jones booked for hauling down the Tranmere striker, it seemed United would have an onerous afternoon.
But Maguire’s spectacular 10th-minute opener instantly turned the tide, the United captain scoring his first goal for the club by finding the top corner from outside the area with a thunderous effort that took a slight deflection and left Scott Davies helpless.
It got even better for the vocal visiting fans three minutes later as Martial played through Dalot, operating further forward than normal, who effortlessly turned past Liam Ridehalgh and coolly slotted across Davies.
Boyhood Manchester City fan Davies had spoken of his wish to put one over on United but was left to collect the ball out of his net once more with barely a quarter of an hour on the clock.
Tranmere’s defenders were again guilty of backing off and allowed Lingard to pick his spot, the England midfielder curling into the net from the edge of the box with aplomb.
Tranmere had battled back from three goals down to grab a draw at Watford in the last round but there was no hint of another improbable comeback as United continued to look menacing any time they went forward and Lingard miscued twice when well placed to increase the lead.
But United’s fourth duly arrived in the 41st minute as Jones’ looping header at the back post found the net for his first goal in nearly six years, while Martial’s deflected effort from the left side of the box put Solskjaer’s side 5-0 ahead on the stroke of half-time.
That was the Frenchman’s final contribution on his 200th United appearance as both he and Nemanja Matic were withdrawn, perhaps with the return leg of the Carabao Cup semi-final against Manchester City on Wednesday in mind.
On came Fred and Tahith Chong, who looked set to get on the scoresheet himself when he poked the ball beyond Davies only to go to ground from the Tranmere custodian’s trailing leg.
Davies was then sent the wrong way as Greenwood casually side-footed home his 56th-minute penalty as Tranmere were left with more than half an hour to try to avoid even further punishment.
Lingard hit the outside of the post with another curling effort while Chong rattled the crossbar from an acute angle but, with the result long beyond doubt, United were largely happy to cruise to the final whistle.