Eric Maxim Choupo-Moting fired Stoke ahead only for Marcus Rashford to level before the break. Romelu Lukaku then fired United in front in the second half but Choupo-Moting's second in the 63rd minute earned Stoke a point.
United had been flying in their opening three games of the season, and Mourinho was quick to blame the international break for disrupting his team's rhythm.
"It was a difficult match for different reasons," Mourinho said on BT Sport. "My players, after the international break, are not the same.
"They weren't performing at their normal level but we were in the game, we fought hard and we were closer to winning than losing.
"Stoke are big, strong and powerful in the air. They fought hard and probably deserved a point. A point is a point. It is not what we came for but a point is a point."
At the end of the match, Mourinho appeared to refuse the offer of a handshake from Mark Hughes, but the Stoke boss shrugged it off.
"I'm not quite sure what happened," Hughes said. "I seem to get involved in handshake issues. It's not the first time, it won't be the last.
"I offered my hand but Jose didn't want to take it. Maybe he was upset about the result. The top managers don't like it when they get the bad results. I have to get used to it sometimes."