Heatly, who had progressed through the morning prelims in sixth place with a score of 346.75 points, knew he had more left in the tank for the afternoon after a wobbly start.
And at the halfway stage of the final, with three dives left to go, he was in fourth place on 190.50 with the leader on 208.95.
The Edinburgh Diving Club member kept his cool and, after nailing a great last dive for a score of 69.70, it meant that he finished on 391.70.
That was enough to see him third behind fellow Great Britain team mate Jack Laugher (gold in 414.60) and Italian Giovanni Tocci of Italy (silver in 401.10).
After adding the medal to his bronze in the same event at the Commonwealth Games earlier in the year the 21-year-old said: “This is very special, I improved massively from the prelim and it paid off.”