Albion jolted Peterborough's automatic promotion hopes with their fifth draw in a row.
The first goalless deadlock at Withdean this season stretched the Seagulls' unbeaten run to 12 matches and dashed the visitors' bid to move up to third.
Peterborough had Jon Cullen sent off on 66 minutes for foul and abusive language.
Andy Clarke missed two good second-half chances for Barry Fry's side, but Albion almost snatched the points in the closing stages.
Captain Paul Rogers headed a cross by substitute Warren Aspinall against the post.Albion made three changes to the team which drew 0-0 at York on Saturday.
Ormerod came in for only his third game since August in place of Mark Walton, who has been suffering from a knee injury.
An ear infection ruled out top scorer Darren Freeman, which meant a recall up front for Ramsay.
There was also a rare opportunity for Thomas. He was given his first start for almost three months at the expense of Ling.
A presentation was made to Hart before kick-off after he was named Nationwide League Player of the Month for March by Sky Television viewers.
The Seagulls' young striker received a glass decanter and bottle of champagne from former World Cup final referee Jack Taylor, now Nationwide's sponsorship executive.
Peterborough, starting much the brighter, should have taken the lead in the fifth minute.
Scott's corner found Cullen completely unmarked at the far post, but his header appeared to be blocked by Ormerod.
It was a good chance for Cullen, signed from Sheffield United for £35,000, to score his third goal in four games.
Albion gradually found their feet on an uneven and patchy Withdean pitch. Skipper Rogers, seeking to lift his season's tally into double figures, headed wide of the far post from a right wing cross by Brooker.
Ramsay was unlucky not to put them in front in the 17th minute. The teenage forward beat goalkeeper Griemink and a couple of defenders to a long ball through the middle by Watson and his lob from 20 yards dipped just over the vacant net.
Ramsay suffered a knock in the process, but he continued after treatment.
Albion were dominating at this stage. Thomas showed up well on a couple of occasions, jinking to the byeline to force a corner and having a shot blocked when Brooker's miscued attempt fell at his feet.
The visitors were at full stretch to keep Albion out, although Lee wasted a breakaway by overhitting a pass to his strike partner Clarke.
Watson, combining well down the right with Brooker at times, tidied up nicely inside his own area when a careless backpass by Rogers put him under pressure from Clarke.
Rogers shot wide after Thomas had another effort charged down.
Lee collected the first caution of the contest in the 33rd minute for a stray elbow on Mayo in the centre circle.
Oatway also had his name taken on 38 minutes for a late challenge on Drury and Scott followed suit two minutes later for a foul on Thomas.
Lee and Clarke, both of whom had Premiership experience with Nottingham Forest and Wimbledon respectively, were being tightly marked by the dependable Cullip and Crosby. Cullip was again having a particularly fine match.
Ormerod, well protected by his back four in the opening 45 minutes, had to be on his toes three minutes into the restart. Albion's keeper tipped over Scott's viciously curling free kick from 25 yards.
Clarke spurned a clean opportunity for Peterborough in the 52nd minute. He combined neatly with Lee before dragging a left foot shot wide from eight yards.
Crosby and Cullen required treatment simultaneously following an accidental collision inside the Albion half.
Cullen was not on the pitch for much longer. He was mysteriously sent off in an off the ball incident midway through the half after losing possession to Thomas.
Referee Michael Jones had no hesitation in producing a red card, presumably for something Cullen said.
Thomas tried to exploit Albion's numerical advantage with a weaving run into the penalty box. His miscue reached Watson, whose shot was saved by Griemink.
Adams took both of his strikers off with 13 minutes left, Aspinall and Ling replacing Hart and Ramsay with Thomas and Brooker forming a new front pairing.
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