With Nick Blackman and Peter Sweeney exchanging strikes before the break, Peter Shirtliff's side seemed destined to secure their first victory of the new League One season when Cullen met Craig Jones' 90th-minute cross.
But, with the net at his mercy, the Bury marksman inexplicably headed over the crossbar.
Blackman, United's most dangerous player, converted debutant Paul Gallagher's pass with a well-executed volley before Sweeney dragged Bury level with a 25-yard free-kick.
Defender Harry Maguire came closest to edging United back in front with a powerful header.
Bury also had a late penalty appeal turned down when Cullen was tackled by Tony McMahon.