The visitors led 26-8 with 15 minutes left after Iain Balshaw scored their fourth try on the hour.
Rory Lawson, Olivier Azam and Olly Morgan had all touched down earlier.
But two tries in seven minutes from James Haskell brought Wasps to within four points before Hart plunged over 20 seconds from the end of normal time.
Danny Cipriani added his third conversion and Wasps held on in injury-time to complete a superb comeback win.
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