The Red Devils took the lead through Anthony Martial in the first half but did not score a second until Marcus Rashford struck deep into second-half stoppage time.
United spurned several opportunities to put the game to bed before then, but Solskjaer was still happy with his team’s performance.
“In the first half I thought we were very good,” the United boss said. “We controlled it, dominated it, scored a nice goal. Second half of course there’s going to be a couple of instances in your box because they’re desperate to get a goal and throwing men forward.
“We could have finished the game off really, we had some great counter-attacking chances. We should have got the second goal earlier but there was some great defending.
“It’s important for us to know that we can grind out results as well, even though when you score one you don’t always win. If we can get clean sheets we’ve got a chance to move up the table.”
“You are always kicking yourself when you don’t take your chances. There were so many great counter-attacking opportunities, when we made wrong decisions or were a little bit too slow to get the finish off, but it was an excellent performance.
“Let’s see what we do against Arsenal [on New Year’s Day. We want to keep that momentum going. Two wins after a disappointing game against Watford. I’m happy with that. [It was a] good response. The boys are learning.”
After Wednesday’s trip to the Emirates Stadium, United face Wolves in the third round of the FA Cup next weekend.