Concerns were raised about Szczesny when he required lengthy treatment after a clash of heads with United's Phil Jones in the first half of the 1-0 Premier League loss at Old Trafford.
The 23-year-old Poland international did not immediately move after hitting the ground following the aerial collision but was soon deemed fit to continue by the club's medical staff.
The incident bore similarities to the controversial clash between Tottenham goalkeeper Hugo Lloris and Everton's Romelu Lukaku the previous week when the Spurs man was allowed to play on despite being knocked out.
Spurs boss Andre Villas-Boas was widely criticised for that but Wenger insisted he would have substituted Szczesny had he been told to do so by medics.
Wenger said: "I listen to the doctor. If he says to me he has to come off, he comes off.
"No matter if we have seven men on the field, I do it.
"You have only one life and you have 60 games per year."
When asked if the doctor had told him Szczesny was fit, Wenger, speaking in his post-match press conference, said: "No, nobody told me anything about him.
"If he said there was a problem I would have taken him off."
United defender Nemanja Vidic did not continue after he collided with team-mate David de Gea in a separate incident just minutes after the game restarted.
The Serbian looked shaken after his head struck the hip of the leaping De Gea and was taken to hospital for examination.
Moyes said: "I hadn't seen the incident from the touchline but I have had a chance to see it (now).
"It's quite a horrific looking one. As far as I know he is okay.
"He is conscious, we will get a scan, but as far as we know, we hope he is going to be okay."
Both clashes will deepen the debate on concussion in sport and the procedures that should be followed when incidences occur.
There had been strong campaigning for improved measures in rugby union before the Lloris incident brought football to the forefront of the discussion.
The charity Headway, which aims to increase awareness of brain injury and its consequences, made headlines last week by condemning Spurs' actions over Lloris.
When asked about the latest two incidents, spokesperson Luke Griggs said: "I don't want to get involved in judging the decision of Szczesny - every situation is unique and different and the medical team have made a call.
"With Vidic, they have taken him off after the medical team deemed him not fit to continue. He was unsteady on his feet, there were more obvious signs."
Lloris, meanwhile, was left out of the side for Spurs' defeat by Newcastle.
A statement from the Wayne Diesel, the club's head of medical services, read: "We've continued to monitor Hugo all week and after discussions, Andre and the medical department have agreed to afford the player a couple more days' rest.
"Hugo is naturally disappointed as he was keen to return to action. However, we feel this decision is in the player's best interest."