Flames' Swedish center scored their penalty in the seventh round, after which Pittsburgh's Brock McGinn's penalty was saved by Swedish goalkeeper Jacob Markström and the match ended 2-1.
Earlier, another Calgary Swede, defender Oliver Kylington, had played until Milan Lucic's 1-0 goal in the second period.
The away team equalized through Jake Guentzel in the third before the match went on to a later goalless extra time and penalty.
In addition to Backlund, Johnny Gaudreau (Calgary) and Kris Letang (Pittsburgh) also scored on their penalties, in the third round.