The Boro boss was left frustrated after what he described as a "bad day at the office" on which the Teessiders were often forced to shoot from long range, despite looking comfortable in possession.
At the other end, Andy Keogh's brace and Liam Trotter's half-volley secured an impressive victory for the Lions, whose defence was breached by Marvin Emnes' solitary strike from the penalty spot just before half-time.
"We lost some bad goals today and we need to improve in both boxes," Mowbray said.
"We've had to make quite a few changes in defence and we haven't been able to get a settled back four yet.
"You can't concede like we did today and expect to take anything off Millwall. We've managed to win here in the last couple of years but not today – you'd have to say that possesion-wise we dominated the game but that doesn't get you any goals or points.
"We had lots of shots but Maik Taylor had a very comfortable day, picking up lots of shots from 20 yards out that weren't struck well.
"Over the 90 minutes we played alright but we didn't have an end product. Up to their 18 yard box I thought we were dominant but we didn't test the keeper, and we didn't get the ball out of our own area."