It looked like curtains for Albion at half-time after Gary Holt put the visitors ahead on 16 minutes.
But Wednesday, not the Seagulls, were booted down into Division Two on Easter Monday after Zamora equalised with a second-half penalty and Stoke won 1-0 at Coventry.
Albion's relegation worries are far from over, however. They must win both their remaining games this season and hope Stoke come to grief.
Seagulls coach Steve Coppell replaced Gary Hart with the injury-prone Paul Kitson while Wednesday's Chris Turner gave starts to Holt, Steve Haslam and Ashley Westwood.
The visitors were first to create an opening, Holt heading Alan Quinn's cross wide after Simon Rodger upended Westwood, handing Wednesday a free kick.
They continued pushing forward and soon turned the pressure into an early lead.
Michael Reddy turned Westwood's cross into the path of Holt, who fired the ball past Beasant.
Reddy later came off injured but his replacement, Lloyd Owusu, provided more problems for the Albion defence as the break approached.
The hosts began the second half in brighter fashion and were rewarded when Holt handled a Kerry Mayo cross in the area on 57 minutes. Zamora kept his nerve to send Kevin Pressman the wrong way.
Albion should have taken the lead five minutes later but Pressman pulled off a stunning save from Kitson's firm header.
With a draw of little use to either side, the game became scrappy with Westwood and Seagulls defender Danny Cullip booked in quick succession for unsporting behaviour.
Turner threw on striker Shefki Kuqi with eight minutes remaining but neither Owls nor Seagulls could find the goal that mattered.
Albion: Beasant, Ingimarsson, Cullip, Blackwell, Watson, Carpenter, Brooker, Rodger, Mayo, Zamora, Kitson. Subs: Hart, Oatway, Jones, Packham, Barrett.
Sheffield Wednesday: Pressman, Westwood, Maddix, Bromby, Smith, Barry-Murphy, McLaren, Haslam, Quinn, Holt, Reddy. Subs: Kuqi, Owusu, Stringer, Wood, Richard Evans.
Referee: K Hill (Hertfordshire)
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