The defender headed in a Dean Cox corner to earn a priceless point for Albion in the relegation scrap at Whaddon Road.
Cheltenham had appeared on course for their first win in five league games with goals from Ian Westlake and Chris Westwood but the Seagulls grabbed a lifeline with a Nicky Forster header after 78 minutes before Hinshelwood powered in a near-post header.
Westlake's cultured left-foot broke the deadlock after seven minutes, drilling in a low shot past the despairing dive of John Sullivan.
Cheltenham doubled their lead in the 58th minute through Westwood, who bundled in a corner from close range, but Ashley Vincent wasted a chance to wrap up the points when he fired wide after good work by Josh Low.
Earlier, Tommy Fraser wasted a golden chance when in the 50th minute he heading unmarked straight at Scott Brown while the Robins keeper made a brilliant point-blank stop to deny Chris Birchall after 70 minutes.