The 28-year-old, who flourished last season on loan at West Ham and earned a recall to the England squad, has two goals in four matches for Manchester this season.
“I think today (Sunday) was a good answer. You know Jesse wants to fight for his place. He wants to be part of a Man United team that he feels is going places,” Solskjaer said.
“He’s really contributing … I’m so happy and pleased for him that he handles the setbacks, he handles the highs and the lows really well.
“He is becoming a grown man, and he’s a very good player.”
United, who are third in the Premier League, will face West Ham again on Wednesday in the League Cup third-round tie at Old Trafford.
Solskjaer said he will make a few changes to the team for that game, with Lingard set to feature against the East London club.