Sanchez has struggled for form and fitness since joining United from the Gunners last January, but he produced a decisive display against his old club at the Emirates Stadium, scoring the first goal in the visitors' 3-1 victory.
Solskjaer has been hugely impressed with the Chile forward's attitude in training since his arrival in December and now seems to think Sanchez is ready to play a leading role in United's season.
"You try to sit him down and speak to him but he's a really hard-working, proud man, and he is really determined to prove his worth," he said. "He has been fantastic since I came. His work rate in training has been brilliant.
"Unfortunately, he has had the injuries, but I think [Friday] is a big, big step forward for him."
However, Solskjaer insists Sanchez is not about to receive any special treatment to help him find consistent top form, simply because he earns a reported £391,000 per week.
6 - In all competitions, Arsenal have lost six matches at the Emirates against Manchester United - more than they have against any other side. Rivalry. pic.twitter.com/zy6yNe0zyf— OptaJoe (@OptaJoe) January 25, 2019
"Everyone gets treated the same, because when you come in and you are looking around the dressing room and there are so many good players you have to prove your worth," he said.
"You don't just live off the price tag. I am not interested in what they have cost or what they are earning. What I am interested in is what they can give the team. But he does really want to be part of it."
United's defeat of Arsenal, which was secured by further goals from Jesse Lingard and Anthony Martial, was their eighth in a row in all competitions since Solskjaer took over from Jose Mourinho in December.
The former striker has now challenged his players not to become complacent as they prepare for Burnley's visit to Old Trafford in the Premier League on Tuesday.
"We're just looking onwards and upwards and to the next game. Tuesday is going to be massive for us. At home in the Premier League, we're expected to win," he added.
"After a big win like this, complacency is maybe our biggest danger. But looking at these players, I don't think that will happen."