The Shots, who had lost to relegation battlers York City last weekend, created an early opening when captain Lorent Tolaj dragged an angled effort wide.
Theo Widdrington’s drive was then turned around the post by Moors keeper Nick Hayes.
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Moors broke the deadlock in the 72nd minute when Tahvon Campbell turned in defender James Clarke’s near-post flick-on for his fifth goal in successive games.
But Aldershot levelled on 89 minutes when 36-year-old former Leicester City, Wolves and Fulham defender Stearman was shown a red card for tripping Tolaj in the box, and Cian Harries coolly converted the resulting spot-kick.
Fourth-placed Moors remain two points ahead of fifth-placed Aldershot with six games left – but narrowed the gap to third-placed Bromley, who lost at home to York.