The Owls made the breakthrough 17 minutes into a scrappy first half when Michael Morrison's long throw was glanced on by Gary Teale and the ball crept agonisingly past goalkeeper Tony Roberts.
The home side were denied an equaliser on the stroke of half-time, with Jon Nurse penalised for a foul on goalkeeper Richard O'Donnell as the pair contested Damien McCrory's deep cross and the ball bounced in off the forward.
Daggers had the first sniff at goal after the break, with Nurse's goalbound header comfortably stopped by O'Donnell.
The home side were level after 79 minutes when referee Steve Rushton controversially awarded a penalty against Darren Potter, who was adjudged to have blocked Darren Currie's cross with his hand. Green made no mistake from the spot.
Wednesday responded immediately, with Potter's fierce low drive forcing a fine save from veteran keeper Roberts. With three minutes remaining, Tommy Miller's effort was blocked by Romain Vincelot from point-blank range as the Daggers held on for a point.