Harry Maguire, Diogo Dalot, Jesse Lingard, Phil Jones, and Anthony Martial all hit in a sparkling first-half performance that saw the Red Devils go into half-time 5-0 up at Prenton Park.
The visitors extended their lead further after the break when Mason Greenwood converted from the penalty spot, and saw out the rest of the half in comfort to advance to the fifth round.
But not everybody was content with the win, with United fans heard singing against the Glazer family that owns the club as well as executive vice-chairman Ed Woodward, the man signalled by many as responsible for their woes in recent years.
Solskjaer vowed to do his best to pull United out of their slump, while taking aim at certain detractors who he believes would like to see the team crash and burn.
“We’ve just got to stay together and keep working. Try to win the next one and the one after that and then we’ll get our players back,” the manager said to BT Sport.
He added to BBC Sport: “You can see everyone wanted us to fail but it doesn’t bother me. I can’t be up and down [emotionally]. I’m happy but won’t relax. Two big games coming up.”
The state of the Prenton Park turf was also a concern prior to kick-off, but it hardly slowed down United as they put in their most prolific first-half performance for almost 20 years.
“A lot has been said and written about this pitch. We played it well. We didn’t want to make it a long-ball fight because that’s not the game we wanted. We put the ball down and scored nice goals,” Solskjaer beamed.
“We have good footballers. They give everything in every game.
“[Tranmere] had the boost of going through, at home, put a couple of balls down channels with corners and set-plays. But when you score two or three goals in quick succession, it’s sit down and enjoy the game and let the players enjoy it.
“Brilliant. It’s important we spread [the goals] through the team. We’ve lost Marcus [Rashford] for a little while. Our two strikers scored – a confidence boost for everyone.”