Carrick unfazed by Rooney support for Man Utd job
Carrick's side stepped up their bid to qualify for next season's Champions League thanks to second-half goals from Casemiro, Matheus Cunha and Benjamin Sesko.
United are in third place in the Premier League, three points ahead of Villa as they chase a top-four finish.
A seventh win in nine games since Carrick took over as interim boss put United six points clear of sixth-placed Chelsea, firmly in the driving seat for a Champions League place.
Rooney said told BBC his former team-mate Carrick should "100 per cent" get the job.