Despite losing leading goal scorer Radim Vrbata to an injury early, Phoenix had a 2-1 lead after Taylor Pyatt and Antoine Vermette scored in the second period.
Coyotes goalie Mike Smith stopped two shots without his stick during a power play in the third period, but Chicago defenseman Brent Seabrook put a rebound past him with 14.2 seconds left in regulation to send the game to overtime.
Hanzal barely nicked Adrian Aucoin's shot on a redirect in overtime to give the Coyotes a 1-0 lead in the best-of-seven series. Game 2 is Saturday night in Arizona.
Jonathan Toews also scored for Chicago in his first game back from a concussion.
Smith stopped 43 shots for the Coyotes.
Phoenix reached the apex of its run in the desert this season, earning its first division title in 33 years as an NHL franchise.
Without an owner for the third straight season and no true star players, the Coyotes got by on grit and great goaltending from Smith, winning their final five games to take the Pacific Division crown. A nice accomplishment, though the carrot for the players was the home-ice advantage that came with it.
For a team that hadn't won a playoff series since 1987 — nine years before they moved from Winnipeg to Arizona — they were happy for any advantage they could get.
They were going to need it against Chicago.
The Blackhawks are fast, skilled and experienced, two years removed from winning the Stanley Cup. They finished with two more wins and four more points than the Coyotes, who earned home ice by winning their division.
Chicago also has two of best lines in hockey and the top one got a big boost for Game 1 with the return of Toews. The Blackhawks' captain missed the final 22 games of the regular season with a concussion and just returned to practice on Monday.
He didn't take long to let the Coyotes know he was feeling good, squeezing a shot past Smith from a tough angle just over 4 minutes into the game — just what the Coyotes didn't want to happen with the fast-moving Blackhawks.
The Coyotes fought back and players from both teams had to be separated in the closing seconds after Hanzal and Chicago's Bryan Bickell were called for roughing penalties.
Phoenix kept up the feistiness to open the second period and broke through 7½ minutes in, when Pyatt lifted a shot past Corey Crawford after a nifty backhand feed by Daymond Langkow. The Coyotes set up the goal by wearing the Blackhawks down, repeatedly keeping the puck in the zone to create an exceptionally long shift.
Vermette put Phoenix up 2-1 late in the period, one-timing a pass from Raffi Torres so hard that it bounced out of the net and Crawford wasn't sure if it went in.
The Coyotes were 33-1 when leading after two periods during the regular season, but Seabrook broke through in the closing seconds when a shot by Toews hit the right post and slipped across the crease right to his stick. Smith didn't know where the puck was and Seabrook flipped it in to force overtime after Crawford had left the ice for an extra attacker.
NOTES: The Coyotes and Blackhawks are meeting for the first time in the playoffs. … Phoenix has killed off 29 straight penalties, dating to the regular season. … Toews' goal was his 16th career in the playoffs. … Chicago's Patrick Kane had an assist on Toews' goal for his 49th career playoff point, moving him into 15th in Blackhawks history.