The England striker scored the equalising goal at Old Trafford after Ki Sung-yueng had given Garry Monk’s side the lead but Gylfi Sigurdsson pounced to earn a 2-1 victory for Swansea in Louis van Gaal's first competitive game as United manager.
Rooney has skippered the team before but has taken the role on a permanent basis this season and is keen to put a losing start to captaincy behind him when United travel to Sunderland in a Sky Super Sunday game on August 24.
"It was a proud moment but a disappointing result," he told MUTV. "It's not one I'll look back on and remember with fond memories.
"You always want to win. It was a nice moment, but I'm disappointed we lost."
Rooney is already looking ahead to the game at the Stadium of Light, where he and his team-mates will aim to make amends.
"We need to prepare right and get this result out of our system. We start our season from there,” he added.
"It's a game we were desperate to win to kick-start our season but we haven't done that. We have to go back during the week and look at how the game went, try to improve and make sure we get it right for the next game.
"It was a disastrous start for us, but we have to move on. We're not happy, the manager's not happy. We'll have to go through what went wrong during the week. We'll do that.
“I thought we didn't play as well together as a team as we can do. We have to put that right."