Kris Boyd opened the scoring after eight minutes and a second goal arrived with 21 minutes remaining as Tony McMahon's free-kick sailed into the far corner, with Leroy Lita claiming the final touch.
Boro are now unbeaten in six league games, but Mowbray said: "I thought we were as poor as we have been in the first half. It was very disappointing.
"We were better in the second half; we moved the ball crisper and passed it better.
"Ultimately, we are where we are and it's about getting three points. We'll take the points and move on, but there's still a long way to go. You can't play as badly as we did in the first half and expect to win most weeks."
Source: PA
Source: PA