Cherries striker Lewis Grabban cancelled out Troy Hewitt's 14th-minute opener for Bury and after David Worrall put the Shakers back in front, the visitors' substitute striker Brett Pitman struck in the 91st minute to make it 10 games unbeaten for Howe since his return from Burnley.
Howe said: "I'm disappointed not to have won the game to be honest. We had enough chances, but again I'm disappointed that we've conceded two poor goals.
"The conditions were very difficult out there, but our approach was good and Brett has shown great character to score there at the end. That's just what he's about. He's harried the defender and taken his goal really well.
"All the players have shown character. If we do want to achieve things this season we've got to keep showing that desire. We don't want to be one of the chasing pack. We've got the quality in the squad to really push on this season now."