The win means United go into their crunch clash with rivals Liverpool on Sunday with a three point lead over the defending champions.
However, there is an area of concern for United, with forward Anthony Martial limping off with what appeared to be a hamstring injury late on in the game.
The France international was replaced by Axel Tuanzebe in the 95th minute after appearing to injure himself when he had a late effort saved by Nick Pope.
Although Solskjaer did not address Martial's fitness in his post-match interview with Sky Sports, the potential for the forward to miss the game against Liverpool will certainly be a concern.
After a slow start to the season, Martial has been directly involved in five goals in his last six league games.
Speaking after the win over Burnley, Solskjaer began to look ahead to Sunday and believes his side are ready to deal with the threat Liverpool pose.
"It's a brilliant position to be in," he told Sky Sports. "Of course we know we're going to the champions, they've had an unbelievable three-and-a-half seasons so we know that's going to be a difficult game for us but we're ready for it.
"I think we couldn't have asked for a better time to play them, because we're in good shape, in good form, we're hungry – that's the big thing, these boys are hungry. They want to get better and that's a test again, a test of character and a test of quality."
Solskjaer then told BBC Sport: "We feel very confident going into games. Away from home we have really done well. We have to believe in ourselves and we do.
"We've done a great job, it's good times. It's a test against the champions who have been fantastic, that's a great test for us."