The Western Conference Central division cellar-dwellers were looking down the barrel of a 15th defeat in 21 matches this season, when the Coyotes went three goals clear in the first period in Glendale.
Antoine Vermette, Michael Stone and Tobias Rieder goals saw the hosts shoot ahead, but the visitors battled their way gamely back.
Gabriel Landeskog struck early in each of the second and third periods, before Matt Duchene equalised with just over 16 minutes of regulation to play.
Daniel Briere scored the overtime winner for the Avalanche, for their seventh win of the season.
Colorado remain bottom of their division, one point behind the Dallas Stars, who also won on Tuesday.
The Stars made it back-to-back victories with a 3-2 regulation triumph over the Edmonton Oilers.
Tyler Seguin's second-period double proved the difference, as the Oilers slumped to a seventh straight loss.
The Nashville Predators and the Ottawa Senators both had come-from-behind, shoot-out wins.
The Predators trailed 2-1 late in the second period against the Los Angeles Kings, but fought back before James Neal was on target in the sixth round of the shoot-out.
Nashville's success, and the St Louis Blues' defeat, saw the former go top of Central, with both sides on 30 points.
St Louis coughed up a 2-0 lead in regulation against the Senators, who prevailed when Bobby Ryan scored in the shoot-out.
Goaltender Ryan Miller earned a shutout for the Vancouver Canucks, who were too good for the New Jersey Devils 2-0.
Evander Kane scored two power-play goals as the Winnipeg Jets trumped the Columbus Blue Jackets 4-2, and the Pacific-division leading Anaheim Ducks held on to beat the Calgary Flames 3-2.