The Red Devils collected their first Premier League away win of the season by beating Norwich 3-1 at Carrow Road.
Solskjaer’s job as the United manager has been under scrutiny following a poor start to the season.
“The one that needs to trust me and the ones that matter, they don’t mind,” Solskjaer told BBC Sport.
United was 15th going into the game against the Canaries, just two points above the relegation zone but climbed up to seventh with their first three points outside Old Trafford
It was only their second game in all competitions in which they have scored more than one goal in a single game after beating Chelsea 4-0 on the opening weekend.
Scott McTominay netted United’s 2,000th Premier League goal, while Marcus Rashford and Anthony Martial were also on the scoresheet, though both players also had first-half penalties saved by Tim Krul.
United held league leaders Liverpool to a 1-1 draw last weekend and beat Partizan Belgrade in the Europa League on Thursday.
Solskjaer said: “This had been a good week for us. It started with a decent performance against Liverpool and now two wins away from home. It gives the players confidence and momentum.
“We have missed Anthony, he is a very important player for us, the way we play and today Marcus Rashford was excellent down the sides and in the middle. A great response to missing the penalty, chasing back and scoring.
“The third was a very good goal and a class finish. It is great for Martial to score a goal like that. We could and should have had a few more goals.”
Asked if he has been hurt by the criticism towards him, Solskjaer replied: “I have been hurt by the results, I don’t know what has been said.
“I do not care about that, anyone can have their opinion and that is none of my business to comment on, I am not hurting because of what someone says about me.”