With scores level in the 81st minute, Lukaku gave the hosts a reprieve when he turned Antonio Valencia's cross over the bar from point-blank range.
The 24-year-old had earlier fired the visitors into a 2-1 lead, only for Eric Maxim Choupo-Moting to quickly restore parity from a corner for his second of the match.
Lukaku's second-half strike boosted his tally to four goals from four Premier League matches, with Mourinho insisting the Belgian's overall package consists of more than just hitting the scoresheet.
"I am happy with Lukaku with or without goals because his contribution is good," Mourinho told the BBC .
"With one more goal we win 3-2 but without the mistake at the corner we win 2-1 and Lukaku's miss does not matter."
4 - No player has scored more goals in their first 4 PL games for @ManUtd than Lukaku (4, same as Saha, van Persie & Ibrahimovic). Impact.— OptaJoe (@OptaJoe) September 9, 2017
Stoke's equaliser came just moments after David de Gea brilliantly denied fellow Spaniard Jese Rodriguez, with Phil Jones culpable for failing to track Choupo-Moting on the resultant corner.
Mourinho bemoaned the costly lapse for denying United a fourth straight league win.
"That was a great save by De Gea, probably the only one he did because in the first half there was only a couple of very slow shots by [Xherdan] Shaqiri," he said.
"After that we could not protect the corner and he was punished. We were punished."