Welbeck scored the opener in United's second Champions League Group A match to give the visitors hope of becoming the only English team to beat the Ukrainian side at the Donbass Arena.
The Red Devils held on until the 76th minute before Vyacheslav Shevchuk grabbed an equaliser.
It denied David Moyes a much-needed three points after a very forgettable start to their Barclays Premier League campaign, yet Welbeck was content to take a share of the spoils.
"I think in the end it was a decent result," said the England international on Sky Sports 4.
"I don't think many teams come here and get them three points and today we would have liked to have got them three points but it wasn't to be.
"I think four points from the first two games is alright, especially considering coming away to Shakhtar and getting the draw.
"We're a bit disappointed, we're Manchester United and we want to win every single game but coming here and getting the point, you've got to be satisfied with it."
He added: "Things aren't going as well as we'd have liked it to in the Premier League but what we've got to do is just gather together and concentrate on our next game and not look too far into the future."
United boss Moyes was pleased his side bounced back from the shock defeat at home against West Brom in the Premier League by picking up a point at a difficult European venue.
"It was a good point in the Champions League coming here, they've got a great record against English teams here," said the Scot.
"The team stood up well tonight, they got a good goal early doors and I thought we looked as if we were going to stop them scoring; we defended very well."
The former Everton boss refused to get carried away, however, and insists they must continue to improve.
"It's a point in the Champions League, it doesn't take us any further just now," he said.
"You have to win other games if we're going to progress further in this competition.
"But I think on paper this looked a hard away tie and we've picked up a good point."
Striker Wayne Rooney was absent from the clash after picking up a shin injury on training, with Moyes saying beforehand that the 27-year-old would have only been a substitute anyway.
Asked whether he will return for the clash with Sunderland on Saturday, Moyes added: "I hope so, it's not that serious. He took a clash of shins with Danny Welbeck and it's just swollen up."