Danny Welbeck's 18th-minute opener had given the visitors the lead before Taison netted late in the second half to take a share of the spoils.
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"The result satisfies both sides," he told reporters. "[Manchester United] are better than us on the break, but I wouldn't say they were in control of the game.
"I am happy we rose to the level of United and even got the better of them in certain respects.
"We were better than United in some respects and it is a pity we did not win, but don't forget we were playing Manchester United."
Shakhtar could have won the game had Taison's late deflected shot on the angle was not diverted over the bar by a fine David de Gea save, and Lucescu said the team followed his pre-match instructions perfectly.
"At half-time, we said to continue playing our way, increase the temp in the last 20 minutes and take some risks," he said.
"That was the case. We delivered many balls into their box and it was a pity we scored just once."