Amid newspaper speculation he could be sacked this weekend, Mourinho's side fell 2-0 behind as early as the 10th minute after goals from Kenedy and Yoshinori Muto fired the visitors - who haven't won in the league this season - into a shock lead.
Needing a win badly, it didn't look like it was coming after trailing Newcastle 2-0 in the opening minutes, but United bounced back and got a late victor from Alexis Sanchez to win 3-2.
Paul Pogba in particular was excellent in the second half as he produced a lovely assist for the Anthony Martial equaliser before the much maligned Alexis Sanchez popped up with the victor in the last few minutes.
Asked if United would be right to sack Mourinho, Sky Sports pundit Neville replied: "No".
It's not the first time that Pogba has shared a moment with the United fans.
The result lifts them only into eighth place in the table, but the impact it could have on Mourinho's future could be hugely important.
Newcastle battled hard for their first win of the season, but their defensive frailties were exposed and their inability to make the most of some half-chances punished.
It was a performance of genuine heart, and one that reminded us Mourinho still has something to give.
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Substitute Juan Mata got United back into the game in the 70th minute with a curling free kick. The decision to remove Bailly was probably a wise one.
There are suggestions that senior United officials met with representatives of Mourinho in London on Friday to discuss the crisis, with the manager backed to turn around the club's fortunes after their worst start to a season in 29-years.
It was a striking remark at a time when his relationship with players is under scrutiny, but Mourinho does not believe confidence is at the root of things.
Mourinho claimed United had overachieved in finishing second last season, 19 points behind champions Manchester City, and said he always expected this campaign to be more hard.
Mourinho knew he needed a win in today's crucial Premier League encounter with his job on the line.
Newcastle will look to get over this result at home to Brighton on October 20.