Patrick Kane also scored for the defending Stanley Cup champion Blackhawks, who won their first game of the season after two straight defeats.
Crawford, making his first start of the season, allowed two early goals before finishing with 32 saves in his ninth career appearance.
"We really needed this win," Hossa said. "We tightened up defensively."
Drew Stafford and Derek Roy each had a goal and assist, and Tim Connolly also scored for the Sabres, who've lost two straight after a season-opening win at Ottawa. Ryan Miller finished with 26 saves.
Crawford helped preserve the win with a sparkling left pad save on Buffalo forward Mike Grier's attempt from the slot with just over 3 minutes to go, and the Blackhawks allowed just one shot during a Sabres power play that began with 2:28 to go and with Miller off for an extra attacker for the majority of the advantage.
"A little panic sets into your head, but you have to try and brush that away," Crawford said of his early struggles. "I just kept battling and stuck with it."
With Chicago up 3-2 after two periods, Hossa notched his second of the game, and third of the season, 3:36 into the third with a wrist shot from the lower left circle that went under Miller's glove.
"They wound up getting one off their skate, but that's what happens when you get pucks on the net," Sabres captain Craig Rivet said. "We would like to take a few of those situations back."
Connolly made it 4-3 less than 2 minutes later when he whipped a shot in from in tight.
Stafford opened the scoring just 14 seconds in when his wrist shot from the inner edge of the left circle eluded Crawford.
Roy made it 2-0 for Buffalo with his fourth of the season 2:29 later.
The Blackhawks responded with four straight goals, starting with Kane cutting the deficit in half with a power-play goal at 7:44 of the first. Leddy tied it after his slap shot went off Buffalo defenseman Shaone Morrisonn's skate near the crease with 16 minutes left in the second.
Hossa gave Chicago its first lead with 1:13 remaining in the second when he went in alone from the blue line and slipped a shot from the slot past Miller.
The game was marred by a scary moment in the first period when Sabres right wing Jason Pominville suffered a concussion after being hit into the boards from behind by Chicago defenseman Niklas Hjalmarsson with 5:42 to go. Hjalmarsson was given a 5-minute major for boarding and a game misconduct. The Sabres were unable to capitalize on the man advantage.
"You never want to see someone laying on the ice like that," Hjalmarsson said. "I thought it was a shoulder-to-shoulder hit. It's good to know that he's up and walking, and hopefully he'll get back as soon as possible."
Material from wire services was used in this report.
1 - 0 BUF
2 - 0 BUF
2 - 1 BUF
2 - 2 Tie
3 - 2 CHI
4 - 2 CHI
4 - 3 CHI
Power Play Stats
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