First-half goals from Harry Maguire, Diogo Dalot, Jesse Lingard, Phil Jones and Anthony Martial ended the game as a contest, and Mason Greenwood added another from the penalty spot in the second period to cap off a satisfactory afternoon.
"It wasn't just the Tranmere people who wanted an upset, there were a lot of others," Solskjaer said.
"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."
When asked about what he was most proud of following the win, Solskjaer said: "I think the players, the way they went about the game on that pitch. We played football the way it should be played and it was an exceptional performance at times because that pitch was very, very difficult to play on."
Despite the thrashing United fans still sang anti-Glazer songs, and there appears to be a divide between the club and supporters at the moment.
Solskjaer responded by saying: "We've just got to keep together, focus on the next one and get a few wins and then the good times will come back."
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