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."
A lapse in concentration saw City fall behind in the 30th minute. McDonald provided an inch-perfect cross from the right for Boyd who had pulled away at the back post.
Boyd, who checked for the offside flag having found himself in acres of space, produced a left-footed cushioned volley, which David James got a hand to but could not keep out.
James flung himself towards the top corner to palm away a curling Williams effort and McDonald ghosted in at the back post but lifted his shot over the bar as Boro looked for a second goal.
A fortuitous leveller arrived just five minutes later.
Adomah's goal-bound shot cannoned off Barry Robson, and Jason Steele, who was already going to his left, could do little about it as the ball nestled in the opposite corner.
Having laid siege to the away side's goal prior to the interval, the hosts then fell behind when Elliott came up with a measured strike just four minutes into the second half.
Elliott showed great technique as he collected the ball and picked out the top right-hand corner of Steele's goal from 20 yards, silencing the expectant home support.
Middlesbrough: Steele, McMahon, McManus, Wheater, Bates, Thomson (Lita 68), Tavares (Kink 77), Robson, Luke Williams (Bailey 68), Boyd, McDonald. Unused substitutes: Coyne, Hines, Miller, Hoyte.
Bristol City: James, McAllister, Carey, Fontaine, Caulker, Rose, Elliott, Johnson (Cisse 87), Adomah (Campbell-Ryce 90), Stead, Haynes (Pitman 90). Unused substitutes: Gerken, Hunt, Clarkson, Stewart.
Referee: C Boyeson (E Yorkshire).
Man of the match: Kris Boyd.