Birmingham, who beat Arsenal in the League Cup Final last month, twice came from behind at St. Andrews but had no reply to Lee's winner which put Bolton into the semi-finals for the first time since 2000 when they lost to Aston Villa on penalties.
They will now return to Wembley for their semi-final next month while Birmingham must spend the remainder of the season fighting their battle against relegation from the Premier League.
Their current skipper Kevin Davies, who scored on Saturday, said before the match: "A lot of fans would like to see us do it for Nat and provide a fitting tribute for such a great man - winning the Cup this year would be perfect."
Birmingham boss Alex McLeish said: "It was a magnificent effort from my team, but we are out the Cup now and know what we have to do in the weeks ahead."
Johan Elmander's powerful left-foot strike put Bolton ahead after 21 minutes of a match which flowed from end to end before Cameron Jerome equalised for the hosts with a low shot that beat Jussi Jaaskelainen at his near post after 38 minutes.
Davies restored Bolton's lead from the penalty spot after he was toppled in a clumsy challenge by Curtis Davies after 66 minutes before Birmingham equalised for a second time with a brilliantly-taken Kevin Phillips goal 14 minutes later.
