Alexandre Lacazette added a second goal on 86 minutes. Substitution Joe Willock sent in a cross and Lacazette's low right-footed shot found the net.
The win sees Arsenal move to seventh place while Wolves remain in sixth.
Arsenal dominated possession in the first-half but apart from a low left-footed shot from Eddie Nketiah that hit the post on 34 minutes, Wolves looked like the more dangerous side.
England youngster Saka, however, opened the scoring against the run of play on 43 minutes with his first Premier League goal.
A ball from Pierre-Emerick Aubameyang sent Kieran Tierney into space on the left. The right-back's resulting cross into the box deflected off Matt Doherty and bounced in front of Saka. The 18-year-old waited for the ball to bounce before sending it whizzing past Rui Patricio in goal.
Nuno Espirito Santo's words were ringing in the ears of the Wolves' players at the start of the second-half as they came out the stronger team from the break.
Increasing the tempo and attacking the Arsenal box, several good chances slid away from the sixth-placed side. Raul Jimenez nearly headed home their opening score on 52 minutes after a cross from Adama Traore.
The Mexican international claimed his diving header had been infringed on by Arsenal's Cedric Soares but the referee waved play on.
Traore nearly made it 1-1 on 64 minutes. After drawing Emiliano Martinez out of his goal, all he had to do was chip the ball over the goalkeeper's head but the forward put too much power on the shot and it went over the bar.
Wolves continued to dominate possession and despite securing several set pieces, they were unable to find the net and Lacazette's goal on 86 minutes sealed their fate.
Arsenal head coach Mikel Arteta said after the game that Lacazette "really needed that goal" after coming off the bench.
"When I asked him to come in, he was ready in three seconds. It was great movement off the ball, a great finish. I am so happy for him. He completely deserves that for the way he trains and plays every day," Arteta said.
The result marked Arsenal's third consecutive win without conceding and Arteta said after the game the team's ability to keep a clean sheet was a welcome development.
"You can see all the teams that are in the title race, that win championships, the amount of clean sheets they have," he added.
"I'm delighted with the performance, the shift that the players put in, the energy that we played, the fight for every single ball from first minute to last minute from every single player that touched the grass."
Also speaking after the match, Saka said it has always been his "dream" to play for Arsenal.
"I am 18 and it has been a dream to play for this club. I have to take each game by game. Wherever the boss wants to put me, I want to play. It is a dream to be here," Saka said.
"It is very big. We know how good Wolves are and how tough it was going to be. We fought for every ball and it is good to build some momentum."