City progressed to Round Four courtesy of a solitary goal from Benjani at The Riverside on Saturday afternoon.
"I am very happy with the win, for the players and the supporters," the City boss said after the 1-0 win against Gordon Strachan's side.
"Benjani is a good player and today he played well and scored and that is important,"
"Middlesbrough played very, very well. They were under pressure in the first 20 minutes, but we scored a goal and it went well.
"I am very, very happy. When you start a new job, it is important that you start well. We have started well, but we must work hard in the next week.
"It is important to improve."