The Coyotes salvaged the finale of a three-game homestand after returning to Arizona on a seven-game winning streak.
The first two skaters for both teams failed to score in the shootout before the Blues' Doug Weight beat Coyotes goalie Mikael Tellqvist. But Shane Doan kept Phoenix alive with a goal against Manny Legace, and then Tellqvist blocked Peter Sejna's shot to set the stage for Nagy's winner.
Ed Jovanovski scored twice and had an assist for Phoenix, and nearly won the game in overtime when his shot went off the crossbar with less than a minute to play.
Jovanovski broke a 2-2 tie with a power-play goal in the waning seconds of the second and then broke a 3-3 tie with another late in the third. But third-period goals by Dennis Wideman and Sejna tied the game again for St. Louis.