The Teessiders slipped out of the play-off places after George Friend headed into his own net and Martyn Woolford converted a simple chance.
Boro staged a fightback after the restart and Curtis Main pulled one back before having a second goal ruled out for offside in stoppage time.
"They were poor, poor goals for us to concede from a defensive point of view," Mowbray said. "We made it very difficult for ourselves.
"We gave Millwall something to hold onto and then we had to try and break them down and we couldn't do that. We had chances to get back into it but we didn't take them.
"We didn't get the luck but we can't keep saying that. It is down to us to win football matches.
"Of course we are very disappointed and of course it has an effect on confidence. And of course it is very frustrating for the supporters here who have seen these results over the past few weeks.
"But do we still believe we can make the play-offs? Of course we do. We know we have what it takes and that we just need to stay strong as a group and keep doing the things we did that won us 11 home games and got us in the top six in the first place."