Kris Boyd gave Boro the lead and they had chances to score three or four more before City levelled through Albert Adomah and then won this npower Championship contest through Marvin Elliott's long-range goal.
Mowbray said: "Of course it was disappointing and of course I would have liked to have started my time as manager here with points on the board but I've got to take the positives from the game.
"We controlled the game for long spells, we pinned them down and we created a lot of chances, and not half-chances but really good opportunities of the sort that Kris Boyd and Scott McDonald have made a career out of putting away.
"I think the shot count was 24 to five in our favour and we created five or six good goalscoring opportunities.
"If we can maintain that kind of work-rate and keep creating chances like that then we will not lose too many games. But the only statistic that counts is that we lost 2-1.
"The concerning thing is that after they scored the second goal - which was a superb strike - we lost our fluency for a while and it was only in the last 15 or 20 minutes that we started to put the pressure on again."
City boss Keith Millen praised his side for handling a difficult atmosphere at the Riverside.
He explained: "I told them there would be a bigger crowd here than usual and a big atmosphere and a lot of expectation on Boro.
"But I told them to enjoy it. These are great games to play in. And the boys did well. We weathered the early storm and really came into our own.
"Boro did well in the first half and they really put us under a lot of pressure but we defended for our lives at times and the keeper (David James) made some good saves.
"But I think Boro ran out of steam and ran out of ideas in the second half and we did well on the counter.
"I think we have turned the corner now. We have done well in the last few games, our workrate is good, we have defended well and we look good on the counter. You can see the confidence coming back."