Solskjaer's side advanced through to the quarter-finals of the FA Cup on Wednesday (AEDT), recording a 1-0 extra-time win over West Ham thanks to Scott McTominay's strike.
It was the Scotland international's seventh goal in 30 appearances for the club this season - as many as he managed in 84 outings in previous campaigns.
United also have a Europa League last-32 tie with Real Sociedad to come later this month and are five points behind leaders Manchester City in the Premier League, albeit having played a game more.
After reaching the semi-finals of three competitions last season, as well as losing at the same stage to City in the EFL Cup last month, Maguire is determined to get over the line this time around.
"Last season, it was disappointing to lose in the semi-finals like we did, but we've got to win trophies," he told MUTV.
"We've got to demand that of each other. We do. We demand that we win trophies and it's vital the team starts winning trophies, for the club and the fans as well."
United's clean sheet against West Ham was their sixth in 10 matches and Maguire felt his side were good value for the win at Old Trafford.
"We got there in the end and we deserved it," he said. "We felt comfortable on the pitch.
"I felt in the first half we played with intensity and dominated the game, played the game in their half and had chances where we should be one, two, three up."
Victory for United was the ideal response to Saturday's 3-3 league draw with Everton, a defensive display Maguire concedes cannot be repeated.
"When we've conceded, we've conceded too many - when we concede one it seems to be two. It was three against Everton. That's not good enough for this club,” he said.
"We were disappointed at the weekend. We felt we deserved the three points but we couldn't feel sorry for ourselves.
"We knew we had to come here tonight and bounce back and get that momentum back on our side ready for Sunday [against West Brom]."
The Red Devils have won their last nine FA Cup matches at Old Trafford - their joint-longest home winning run in the competition, equalling their nine in a row between 1908 and 1912. Maguire is hoping they will get to play at their own ground again when the draw for the last eight is made on Thursday.
"Yes, of course. We love playing at this ground, and we've got to make it tough for people to come here and get results," he said.
"We need to get better and we seem to have improved a little of late. But no, obviously, we want a home game.
"We fancy our chances against anyone. We just need to look after ourselves, perform well and assure that we start winning trophies at this club."