A Robin van Persie hat-trick guided the Red Devils to a 3-0 triumph at Old Trafford in the Round of 16 second leg allowing the side to progress to the quarter-finals 3-2 on aggregate.
The victory comes at an important time for David Moyes' United after they dropped 12 points behind fourth position in the Premier League and according to Rooney, it could become the catalyst for a strong finish to the campaign.
"Of course we did," he told Sky Sports when asked if United owed their manager a positive result.
"We owed ourselves a good result. The fans, the manager, everyone.
"This is a great result and hopefully this will kickstart our season and we can start something really special."
He added: "The fans were fantastic against Liverpool and again tonight they were brilliant and they deserve this. It was a big result, a big performance and we're delighted for them."
United return to domestic action when they head to Upton Park to take on 12th-placed West Ham United on Saturday.
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