The Swiss champions were put on the backfoot after two quickfire goals by Danny Welbeck early in the first-half.
But they stunned the Old Trafford crowd by staging a remarkable comeback through Fabian Frei and Alexander Frei (2).
Fink's side were then denied a famous victory against the English champions in injury time when Ashley Young equalized for United.
Speaking after the match, the German coach admitted he is pleased despite missing out on three points.
"I am pleased, even though we conceded the equaliser right before full time. The team has shown what kind of morale we have and what potential we have to play football," he was quoted as saying in UEFA.com.
"We did not want to hide, we wanted to capitalise on those small weaknesses in United's defence - and that's what we did in the end."
Fink went on to reveal his side's philosophy during the match and once again expressed his delight in the result.
"We had our best periods and created the most chances when we played the ball through the team and showed patience. If you play hastily, you run into difficulties.
"That was our problem in the first half, despite playing well. A 3-3 here in the Theatre of Dreams is a nice result, but if I had dared to dream, I would have dreamt of a win. The fact that we were not very far from achieving that makes me incredibly happy.
The draw puts Basel top of Group C while United are third.