The full-back was excellent for much of a gripping Pool 1 contest, scoring a try and 10 points from the tee to put the Premiership side 20-12 up early in the second half.
Burns went from potential hero to zero in the closing stages, though, as Bath strived to snatch victory after Toulouse fought back to lead by two points, with Sofiane Guitoune scoring his second try of the game.
The Top 14 side were given a let-off when Burns struck a post with a penalty from close range, but things went from bad to worse for the 28-year-old five minutes from time.
Burns burst through for what seemed to be a decisive try, but after blowing a kiss to the crowd and tapping his chest, a broad smile was wiped off his face when Maxime Medard flicked the ball out of his hand before he could touch down under the posts.
A distraught Burns was hauled off moments later and Toulouse held on for a 22-20 win, Medard named man of the match after also scoring a first-half try.
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