Both sides would have gone third had they picked up a victory at Westleigh Park, but as it was a 2-2 draw lifted Dulwich into fourth spot and Hawks into sixth.
Hawks twice came from behind to stretch their unbeaten league run to six games, with all four goals coming in the first half.
It was again end to end stuff in the second period, though clear cut chances were far fewer as Dulwich comfortably held on despite playing the final 13 minutes a man short after sub Chike Kandi was sent off for retaliation.
Hawks boss Doswell named an unchanged starting XI from the weekend’s FA Trophy loss at Slough which, due to injuries, was no surprise.
Dulwich started on the front foot with Danny Mills - once a target for Lee Bradbury during his time as Hawks boss - shooting wide on three minutes and heading wide two minutes later.
It was Mills who duly opened the scoring on 12 minutes, latching onto a cross from left wing-back Andre Blackman and steering a close range shot past Will Mannion.
Hawks levelled on 24 minutes through the collector’s item of a Baggie header - the shortest player in the home team getting in front of a defender and flicking a Roberts cross over stranded keeper Charlie Grainger.
Hawks now had their tails up, and looked the more likely to grab a second goal. But just before the interval, Joe Newton was penalised for a foul on Jack Holland - Hawks having missed the chance to clear a set piece - and Green casually converted the spot-kick.
Shortly after the fourth official had held up the board to indicate two added minutes, Baggie levelled for a second time.
Billy Clifford did the hard work, gaining possession and playing the winger in. Baggie, on the right side of the penalty area, had time to take a touch before rifling a shot past Grainger.
Hawks started the second half on top, Clifford seeing a half-volley pushed behind for a corner by Grainger.
Shortly after, skipper Joe Oastler got on the end of a far post corner but couldn’t direct a header on target.
Hawks would not get many clearer chances in a fairly even second period which swung from end to end.
Tyrone Sterling’s shot was deflected over for a Dulwich corner, while at the other end Baggie advanced into the penalty area but saw a shot blocked by a defender.
Dulwich made their first sub on 75 minutes, introducing Kandi. But eight minutes later Kandi was trudging off after receiving a red card for retaliation on Clifford.
Tommy Wright was closest to a Hawks winner, Grainger saving a close range shot, but overall the draw was the fairest result.
There was six minutes of added time, but Hawks never really threatened a winner which would have been harsh on Dulwich.
Hawks: Mannion, Passley, Newton, Oastler, Rooney, McCarthy, Gobern, Clifford, Baggie, Roberts, Wright.
Dulwich: Grainger, Taylor, Sterling, Blackman, Timlin, Holland, Higgs, McGregor, Mills (Kandi, 75), McQueen (Barnum-Bobb, 88), Green (Harris-Sealy, 79).