Bledsoe held a share of the record with Hall of Famer Terry Bradshaw before finding Henry as a safety valve 1:58 into overtime, giving the Bills (2-2) their second victory in three overtime games this season.
"We'll take them how they come," Bills receiver Eric Moulds said. "This is a tough league and a tough opponent we played today."
Mike Hollis had a chance to win it for Buffalo in regulation but had a 39-yard field goal blocked with 25 seconds remaining.
"Another tight one," Bills coach Gregg Williams said. "Proud of the guys and how they were able to bounce back from the (blocked) field goal in regualtion."
A three-time Pro Bowl selection who was acquired from New England in April, Bledsoe entered the year as one of five NFL quarterbacks with two touchdown passes in overtime. He had a 48-yarder two weeks ago for a 45-39 win over Minnesota.
"It was a great block downfield by Price," Henry said. "I just read it and I saw the end zone and I just knew I was going to get that. I was just happy to get the win."
Henry fumbled for the fourth time in as many games in the fourth quarter and the Bears returned it for a touchdown. But he made up for that by taking Bledsoe's pass, finding a seem on the right side of the field and dashing down the sideline for the winning score.
"We know the situation," Bledsoe said. "He's got to hang on to the football.
"He's a guy we all believe in. We're counting on him. He's an outstanding running back. We're going to keep putting him on the field. I was really happy for him when he made that decisive play."
The winning drive took just two plays. Bledsoe started it with a 30-yard pass to Eric Moulds, who leaped over the back of a defender for one of his eight catches.
Bledsoe finished 28 of 36 for 328 yards and four touchdowns. He completed all 15 of his pass attempts on Buffalo's four touchdown drives, also finding Peerless Price, Moulds and tight end Dave Moore.
Jim Miller threw two touchdown passes for the Bears (2-2), who dropped their second straight after a 2-0 start. All four of their games have been decided by six points or fewer, a trend that began last season.
Dating to the 2000 season, Chicago had won 12 straight games decided by seven points or less before being beaten by New Orleans last week.
"We feel we have a good team here," Miller said. "We want to be in close games like that, and today we fell short."
Entering the week, three teams had each of their games decided by six points or less -- the Bears, Bills and New York Giants.
Moulds had 119 yards receiving for Buffalo, which had 419 yards of total offense to 241 for the Bears. Chicago had just 51 yards rushing.
"We knew it would be very difficult to play here," Bears Coach Dick Jauron said. "We got to come in and steal a win. We almost did it."
The Bears had possession first in overtime but went three-and-out. Brad Maynard then got off a 31-yard punt and Charlie Rogers called for a fair catch at Buffalo's 44.
On first down, Bledsoe lofted a ball down the left sideline that Moulds caught while reaching over cornerback Todd McMillon.
"I've said it often and I'll continue to say it -- our receivers, one-on-one on pass coverage, I feel they're going to win every time," Bledsoe said.
Bledsoe dropped back on the next play and was nearly caught from behind by rookie defensive end Alex Brown, but he rolled right and found Henry for the decisive touchdown.
"Everybody on the sidelines knew we were going to win this game," Bills defensive end Chidi Ahanotu said. "We knew their offense wasn't doing anything against us, and when Drew got out there, their defense would not stop him."
Bledsoe threw his first touchdown pass to Price to open the scoring in the first quarter. Henry fumbled late in the period and the Bears turned it into a score as linebacker Rosevelt Colvin lateraled to safety Mike Brown for a 62-yard return.
"I got caught switching the ball again and it came out," Henry said. "It's something I have to work on. It was rough, but not one time did I doubt my skills or my toughness."
When linebacker Warrick Holdman stripped the ball and Colvin scooped it up, it marked the 22nd straight game the Bears forced a turnover.
"I think that's what our focus is every week, to create turnovers, to strip the ball out," said Colvin, who also had two sacks.
The Bills scored the next 10 points for a 17-7 lead as Hollis kicked a 48-yard field goal and Bledsoe found Moulds for a four-yard touchdown.
Miller got Chicago within a field goal when he found David Terrell from five yards. To celebrate, Terrell simulated driving a car, rolling down an imaginary window while steering with his back on the goalpost support.
The teams exchanged field goals in the third quarter before Paul Edinger's 24-yarder tied it at 20-20 with 12:30 remaining in regulation.
Officials had ruled a touchdown after Terrell came down with the ball on the left side of the end zone. But the Bills challenged and the call was overturned when officials ruled Terrell had only one foot inbounds.
Bledsoe later capped a 78-yard drive with a one-yard touchdown pass to Moore, but Miller erased Chicago's third deficit with a three-yarder to John Davis, tying it at 27-27 with 2:46 left in the fourth quarter.
Miller was 19 of 31 for 188 yards. Marty Booker had four catches for 82 yards.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys