The Red Devils were on the cusp of defeat until substitute Wayne Rooney made history, bending a 94th-minute free-kick into the far top corner to become the club's all-time highest goalscorer on 250.
That earned United a precious point to extend their unbeaten run in all competitions to 17 games, having trailed to Juan Mata's 19th-minute own goal - the Spaniard's outstretched leg taking Erik Pieters' low cross beyond David De Gea.
But despite the nature of the last-gasp Premier League draw, the United manager came away disappointed - his side having had 65 per-cent possession and 25 shots on an afternoon they dominated at the Bet365 Stadium.
Draw was not a good result, declares downbeat Mourinho
The Portuguese boss told reporters in his post-match press conference: "I think you should speak about Wayne's record today and tomorrow, the next 24 hours, and then finish. Let him be again a normal guy.
"He is a legend in the history of our club and a legend in the history of English football, but the record is in his pocket and now I just want to let him be a normal player and try to score more goals for us.
"It was a great goal. In a team where you lose incredible goals week after week, to get a phenomenal goal like he did is even more astonishing because sometimes I'm on the bench and I'm tired of seeing my team play so well, create so many chances and it's always dramatic to score a goal.
"I'm saying for a long, long time that one day we will score every chance of the match and somebody will go hope with six or seven in the basket but the reality is that day never arrives and we always struggle so much to score goals.
"We hit the post, the goalkeeper saves, we miss a chance, and in the end we risk to lose the match by scoring in our own goal.
"You have to wait for minute 90-something to score the equaliser in spite of keeping a phenomenal unbeaten run and showing amazing character by the team and show an amazing empathy between the team and the fans.
"It is not a good result. I think we lost two points."
Finishing the 'only aspect' Mourinho would change about United's performance
Asked if he feared there wasn't going to be a late goal to earn a point, Mourinho continued: "Of course, of course.
"We were on the bench, you see your team missing chances and you see your team scoring in its own goal, then you see your team missing a chance with an open goal and then you miss more chances and the goalkeeper saves, and the ball goes to the post. Obviously [we were worried].
"During the season it happened to us not once or twice but so many times, to have the chances and not to score goals.
"[It was a] Very difficult match, I have to say that Stoke started strong and after that they defended with absolutely everything with great emotion and aggression, good aggression not negative.
"They did everything to try to defend their goal, but in spite of that, it is a match that we have to win.
On whether United's below-par finishing is the only thing he would change, Mourinho added: "We defended phenomenally well, we kept the opponent very far from our goal.
"We had control, we play, we create, we didn't score goals. I think it's the only aspect that I would change.
"In reality, when the game is over, it's the only thing that really matters - how many goals you score, how many goals you concede."
Today we showed why we draw so many games, says frustrated Mourinho
Rooney's stunning free-kick, his first goal from off the bench since the 2010 League Cup final, came 215 games quicker than Sir Bobby Charlton's previous record.
And Mourinho said that Rooney's wait to break the long-standing record has dragged on for too long, when asked to expand on his previous comments about letting him be a normal guy again.
He continued: "[It has gone on] For too long. Since the beginning of the season. I know it's an amazing record and it's something not to forget, it is an amazing achievement.
"It's great because I think tomorrow he gets your award, from the football media writers. He gets that as an achievement for his career. It comes in the most beautiful moment.
"You couldn't choose the Sunday better than this one, because it goes with the record. Amazing moment for him. But now life goes on."
Mourinho had set himself the target of challenging for the title at the start of this season but they could find themselves as many as 14 points behind leaders Chelsea at the end of this weekend.
Despite that, and having lost further ground in the top-four race with this result, Mourinho insisted his team are always improving.
"I set the target because there is nothing else you can do at Manchester United," Mourinho continued. "I expect them to set the targets really, really high.
"For some reason, United in the last three seasons didn't win the title. For some reason. In football it is very difficult and we are improving in many aspects.
"For me, the biggest improvement is the fact that the next match, we want to win. It doesn't matter the opponent, it doesn't matter the competition. The next match we want to win and we go to every match with that mentality.
"Obviously you don't win all the time and in our case, in the Premier League, we drew too many matches and today you show why we drew so many matches.
"The opponent, they score without even one single chance. And for us, to score a goal, you have to create six or seven chances so it's hard."
Mourinho also declared that defender Marcos Rojo was not well enough to feature, the club insisting he missed out through illness although he was spotted among the travelling United supporters.
With winger Anthony Martial absent from the 18 altogether, he added that the Frenchman was not injured and simply "was not selected."