Williams enjoyed a superb afternoon, and he received a standing ovation before winning the Man of the Match award.
Solskjaer's selection issues
Left-back has been a problem position for United over the past few years, and Solskjaer will hope that Williams can be the long term solution.
Luke Shaw has been out of action with a hamstring injury since United's 2-1 defeat to Crystal Palace in August.
The 24-year-old has struggled to keep fit since signing for United in 2014, while Ashley Young turned 34 earlier this year.
Williams has only made seven first team appearances for United, but he has already shown his potential.
Against Brighton, he won four of his five tackles, registered a pass completion rate of 88 per cent and made five ball recoveries.
Overall it was a solid performance from the academy graduate, and he could become an accomplished full-back in the coming years.
"He's really, really impressed me."
Solskjaer has put his faith in youth this season, and he was particularly impressed by Williams' performance against Brighton.
"Brandon was excellent. He’s played well before, but first start at Old Trafford in a Premier League game, you don’t know how he’s going to react," said the Norwegian.
"I told him on Thursday just look forward to this one, I thought it was more fair to him when he’s playing his first game that he didn’t play on Thursday, so excellent performance.
"He’s done nothing wrong since he’s come in and making it hard for me to leave him out. He’s really, really impressed me, I have to say."
Young and Shaw will both be available after the international break but they may struggle to get back into the starting line-up.