The Red Devils were held to a 1-1 draw by Turkish giants Fenerbahce away from home on Thursday.
While Onana had to make some vital saves, he also felt they did have chances to claim all the points.
He told TNT Sports: “It is a big disappointment for me and the fans, for the club, as a Manchester United player you have an obligation to try to win. It is what we tried to do today, but we also have to recognise it is a tough place to win.
"We didn’t win, but at least we didn’t lose, Not happy of course, but we move on, take responsibility and go again.
“We had more control than them in the second half and more chances. We conceded an early goal in the second half, we had to show character and I think that is what we did. We didn’t win but I thought we were better than them."
On his superb double save, he added: “The saves are not important, the most important thing is the victory for the team. It is nice if I make those kinds of saves and the team wins. It is a big disappointment.”