The Chile star opened the scoring against his former club, setting United on the way to a 3-1 win.
Ole Gunnar Solskjaer has now won eight consecutive games since taking over from Jose Mourinho.
Sanchez has struggled to justify his huge wages since arriving at Old Trafford last January.
But the ex-Barcelona star is the latest Manchester United player to rediscover his form under Solskjaer.
And the Norwegian manager thinks Sanchez’s performance against Arsenal could kickstart his 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 about the 30-year-old striker.
“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.”
Solskjaer denied that Sanchez is treated any differently from his team-mates due to his mega-money contract.
“Everyone gets treated the same,” he continued.
“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.
“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.”
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,” he added.
“Tuesday is going to be massive for us. At home in the Premier League, we’re expected to win.”