Collison's last two foul shots came with 3 seconds remaining and the Mavericks survived a missed 3-point attempt by Shannon Brown at the buzzer to win on the road for only the third time in 11 tries this season.
Phoenix lost its fifth in a row.
O.J. Mayo led the Mavericks with 22 points, five rebounds and five assists. Collison and Brandon Wright both came off the bench to score 16 each and Chris Kaman delivered 15 points to go with seven rebounds.
The Mavericks had lost four of their previous five and on Saturday against Houston will attempt to win back-to-back games for the first time in a month.
Goran Dragic and Markieff Morris both had 15 points for the Suns with Morris pulling down 17 rebounds.
Man behind Doritos Locos Tacos passed away on Thanksgiving
Caroline Berg Eriksen: Soccer player's wife triggers debate with post-birth selfie