There were just four minutes of normal time remaining when Reid supplied the cross for Connell to score with a glancing header, seven minutes after Stanley had levelled through Rommy Boco.
Bradford deservedly went in front in the 24th minute courtesy of a goal from Garry Thompson, who scored with a superb right-footed shot into the far corner after Rory McArdle's long free-kick had been headed into his path by striker James Hanson.
Accrington, without a win in seven matches, were a more dangerous side in the second half and equalised in the 79th minute, despite going down to 10 men when Dean Winnard was sent off for a second bookable offence following a foul on substitute striker Nahki Wells.
The home defence failed to clear a free-kick from Accrington skipper Luke Joyce and Benin international midfielder Boco stabbed the ball into the net from close range, but Connell hit back to win the game for Bradford.
