The Stars ended a four-game losing streak, which had been the longest current stretch of futility in the NHL heading into Friday's action.
Hanzal's goal came 45 seconds into the second period and after Michael Ryder put the Stars back in front, Hanzal scored again with 1:34 remaining in the middle frame.
That was all the scoring until the shootout, in which Vrbata and the Stars' Jamie Benn both produced a goal during the opening three rounds.
After the 19-year NHL veteran Jagr scored for Dallas, Oliver Ekman-Larsson was turned aside by goaltender Kari Lehtonen.
Lehtonen made 31 saves during regulation and the overtime period. Mike Smith stopped 18-of-21 shots.