A goal in each half from Darren Pratley saw off the misfiring Robins, who have now failed to score in their last five games and are in danger of slipping back in to the Championship relegation zone.
Millen was slow to attend his post-match press conference at Ashton Gate, having clearly conducted a dressing room inquest.
When he did arrive, he said: "We are in a fight now and we have to go to Preston on Saturday and get a result.
"Confidence was low, especially at home, and you could see the self-belief drain out of the players when Swansea scored the first goal.
"I told them at half-time that we couldn't play any worse and made a couple of changes. There was a response after the break and, for a time, we looked like getting back in to the game. Then we conceded a really soft goal from a corner and, in the circumstances, that was bitterly disappointing."
Source: PA
Source: PA