Danny Welbeck's 18th-minute opener looked as though it would be enough for the Red Devils to take all three points from their UEFA Champions League Group A clash, only for Taison to lash home a leveller 14 minutes from time.
And Moyes, who has come under increasing pressure following a poor start in the Premier League after replacing Sir Alex Ferguson, was impressed by the resilience of his side at the intimidating Donbass Arena.
The Scot told Sky Sports: "The team stood up well tonight, we got a good goal and I thought we looked as if we were going to stop them scoring tonight, we defended very well.
"We said at half-time we needed to keep the ball better, but we actually gave it away more in the second half, so I was disappointed we didn't pass it better.
"We carved out a really good chance and Robin (Van Persie) was a bit unlucky not to get a second goal and I think if we'd got that we would have seen it out quite easily.
"I think their goal goes through Michael Carrick's legs and then goes through one of their players' legs before it reaches Vida (Nemanja Vidic) and they get a ricochet which gets them the goal, so we were really unlucky to lose a goal.
"Although it is a point in the Champions League, it doesn't take us any further just now. We're going to have to win other games if we want to progress in the competition.
"On paper this looked a hard away tie and we have picked up a good point."
United were missing in-form striker Wayne Rooney after he suffered a shin injury in training ahead of the game in Ukraine, but Moyes is hopeful the England international will be fit to return against Sunderland on Saturday.
When asked if Rooney would feature on Wearside, he added: "I hope so, it's not that serious. He took a clash with Danny Welbeck and it's just swollen up."
And Moyes also took the time to salute the evergreen Ryan Giggs, who made Champions League history with his 145th appearance in the competition when he stepped off the bench to replace Marouane Fellaini in the second half.
The United boss said: "It was fantastic. I was telling him before the game he was going to be subbed on, he was going to come on off the bench. What a record, and I've got to say, what a player."