Luke Shaw's first-half own goal - which occurred when the full-back deflected Luis Suarez's header into the net - proved to be the difference between the two teams in a match that saw Barca muster just three shots on target across the 90 minutes.
That was three more than Ole Gunnar Solskjaer's side managed, though, and United have it all to do in the return fixture at Camp Nou next week.
Alba admitted the Spanish club would have liked a bigger advantage to take back to Spain but was pleased they kept United at bay at the other end.
"It was a balanced game with a couple of well-organised sides," he said.
"This is the Champions League. We tried not to make mistakes and the team played very well defensively.
"On the attack, we also played some good football. The result is a bit tight but scoring away and not conceding is good for us.
"We controlled every single aspect of the game perfectly. We had most of the ball, we played well in the air, where they are so powerful, we also kept our balance to avoid their counter-attacks.
"We were fluent in the build-up. We can be happy as it is not easy to win here. We have to play the second leg but now we have the lead."
It was an unusually quiet night for Lionel Messi in Manchester, though the Argentina forward did still provide the cross for Suarez which led to Barca's goal.
He was left bruised and bloodied following a collision with Chris Smalling when the United defender attempted to win the ball, and coach Ernesto Valverde confirmed he will be examined further on Thursday.
"You guys all have seen him, what happened to him," Alba added. "Hopefully it won't be a big issue."