Jackett had led Millwall through an unbeaten eight-match run before the clash at the New Den, but the Premier League visitors were already four goals to the good and set for the fourth round when Danny Schofield netted an injury-time consolation.
"We're disappointed with the result," said Jackett.
"We wanted to try and cause an upset. Birmingham were clinical and that made the difference."
The Lions boss said his side had been hampered by the absence of Jason Puncheon, who returned to Southampton this week after scoring five goals in a month-long loan spell, and admits he wants the midfielder back at the New Den.
"Jason scored five goals in seven matches and he is under contract at Southampton," added Jackett. "We are keen on Jason. We'll see what comes about in terms of who's out there. He is still a target."