A 39th-minute goal from Scott McDonald proved good enough to earn Boro a 1-0 victory at Sunderland in their fourth round tie at the Stadium of Light.
"That's our fourth Capital One Cup game away from home so whoever we get and when the night comes we'll apply ourselves as we did tonight," said Mowbray.
"I am delighted for the team and the supporters. It was a decent performance.
"Cup competition is about getting through to the next round and we managed to do that. I think there will be a lot of happy people on Teesside."
Sunderland are struggling to find the net at the moment but Mowbray expects boss Martin O'Neill to revive their fortunes.
"I have total respect for Martin and his team," he said. "I'm sure he's scratching his head as to why they've not scored as many goals as he'd want.
"They've got a lot of attacking prowess and a lot of players who can score goals. I'm sure Martin will find the answer to the conundrum he's got at the moment and find more goals."