The Flames lost their ninth straight game, equaling their longest losing streak in 24 years. Calgary's last two setbacks have been in shootouts.
After Calgary fell behind 2-0, Mark Giordano and Mikael Backlund scored to give the Flames a chance to end their long skid.
Both Olli Jokinen and Backlund, however, missed the net in the shootout. That left the Coyotes with their second straight win and gave them their 20th home victory of the season.
Ilya Bryzgalov made 22 saves and has won five of his last seven starts.