A good first half performance saw the visitors go in level, having seen Callum Taylor receive a straight red card after jumping into a challenge.
But they fell behind in the 48th minute when Bird beat Jack Thomas in a challenge on the wing before moving into the box and firing high into the net.
Blues went close to levelling three minutes later when Dipo Akinyemi fired a powerful strike that keeper Horsell pushed away but into his own defender and he just managed to prevent an own goal.
The high-flying hosts doubled their lead in the 58th minute when Smith scrambled the ball home, and although Jamie Cureton hit the bar, the Bulls closed out the win to send the majority of the 1,722 crowd home happy.
Full report to follow