After Chelsea twice hit the post, the Red Devils were lucky to lead 1-0 at halftime through Marcus Rashford's penalty, BBC Sport reported.
But United improved after the break as Frank Lampard's first match as Chelsea boss ended in a day to forget.
"We know we're not anywhere near the finished article," Solskjaer said.
"We are working on improving as a team and finding a style that suits us a team. We still need time but it is coming.
"Winning is a habit. We are lucky to go in 1-0 up but in the second half our nerves settled, our confidence grew."
Solskjaer's team finished sixth in the Premier League last season, 32 points adrift of champion Manchester City and five behind Tottenham in the race for Champions League qualification.
Few outside of Old Trafford have predicted United making a title challenge this season, with many believing a top-four finish would represent satisfactory progress for the 20-time English champion.
Some supporters are not convinced United did enough in the summer transfer market, having signed three players – Leicester defender Harry Maguire, Crystal Palace right-back Aaron Wan-Bissaka and Swansea winger Daniel James – and letting striker Romelu Lukaku go to Inter Milan.
England international Maguire, now the world's most expensive defender after joining for £80 million last week, was named the man of the match after a commanding display.
Wan-Bissaka, a £50 million signing, also looked assured on his first competitive start, while Wales international James marked his debut with a goal seven minutes after coming on as a substitute.
"Harry has done exactly what we've signed him for, his presence in both boxes, his calmness and composure on the ball," Solskjaer, who replaced Jose Mourinho on a temporary basis in December before being given the job permanently in March, added.
"Aaron is so aggressive, he doesn't want to get beat, he's a proper Manchester United defender.
"Daniel made a great impact and it is the best feeling ever scoring a goal in front of the Stretford End, you can’t beat that."
'Question mark' over future
Meanwhile, Manchester United midfielder Paul Pogba said there is still a “question mark” over his future following his victory on Sunday.
The France international said he wanted to leave the club during the close season and was linked with a potential move to Real Madrid or Juventus, Reuters reported.
Solskjaer praised Pogba’s professionalism after he set up United’s third and fourth goals against Chelsea, but the midfielder said “only time will tell” if he would stay in England.
“I always feel good when I play football. I do the things I like, plus it’s my job. I give my all every time I go out on the pitch,” Pogba told French radio station RMC.
“I’m aware that things have been said. Only time will tell. There’s always that question mark.
“I’m at Manchester. I have fun with my team mates. I always want to win matches and I always give my best when I’m on the pitch.”
The Premier League transfer window closed on Thursday, a day before the start of the season, but top-flight clubs in England are still allowed to sell players until the European window closes on September 2.