Solskjaer's side had to work hard for their last-four place as Harry Maguire won it late 2-1 in extra time at Carrow Road.
Odion Ighalo's opener was cancelled out by Norwich's Todd Cantwell, but Canaries defender Timm Klose was sent off for hauling down Ighalo in the 89th minute.
United gradually wore Norwich down and England defender Maguire pounced to extend their unbeaten run to 14 games in all competitions.
United, who won their 12th and most recent FA Cup in 2016, remain on course to reach the final for the second time in three years.
"I'm delighted to be in the last four. Many of these players did not play the last game and now have 90 or 120 minutes in their legs," Solskjaer said.