Reinhart had two goals and two assists, Carter Verhaeghe got the go-ahead goal early in the third period and the Panthers withstood two highlight-reel scores from Bedard to beat the Blackhawks 4-3 on Sunday.
Oliver Ekman-Larsson had a goal and an assist, while Matthew Tkachuk and Evan Rodrigues each had a pair of assists for Florida. The Panthers have won a season-high four straight and are 9-2-1 in their last 12 games.
"Any time you can get home and pile some points together, it's key," Reinhart said.
Bedard - who has nine goals in his first 13 games - became the youngest player to score twice in a game against Florida, doing so three days after he became the youngest player to have a two-goal game against Tampa Bay.
"It's nice to see him rolling and feeling good about it," Chicago forward Nick Foligno said. "You can see the game's starting to slow down for him, which is scary."
Sergei Bobrovsky stopped 24 shots for the Panthers, who were leaving after the game for a 5 1/2-hour flight to San Jose. A three-game West Coast swing starts Tuesday.
There have been three instances of a player having multiple points in four straight games so far in this NHL season. Detroit's Dylan Larkin has one - and now, Reinhart has the other two such streaks. He has 10 points in his last four outings, and his four-point game tied a career best.
"Happy for him," Panthers coach Paul Maurice said. "He's worked hard to get himself to this point."
The Panthers never trailed, with Ekman-Larsson opening the scoring 39 seconds into the contest - Florida's fastest goal to open a game in just over a year.
Reinhart's goals gave Florida leads of 2-1 and 3-2. Both times, Bedard would net the equalizer.
The 18-year-old scored his first of the afternoon late in the opening period, taking the puck off the stick of Florida's Kevin Stenlund near the goal line to the left of Bobrovsky, seeing an opening and flicking one over the left shoulder of the Panthers' goalie from a tight angle.
Bedard's second goal, midway through the second, came after picking up the puck along the left-wing boards between the blue line and center ice, skating in and beating Bobrovsky low to the glove side.
"He's an elite player," Rodrigues said, "especially in transition."
Both scores were unassisted. The Blackhawks were Bedard's helpers in other ways.
When Florida's Dmitry Kulikov - an NHL rookie when Bedard was 4 - delivered a hard-but-clean check on the young Chicago star late in the second, Foligno immediately took offense. He went at Kulikov and was penalized for hooking and roughing, giving Florida a 4-minute power play.
"I just didn't like the hit," Foligno said. "You can't let one of your best players get hit like that, right?"
Chicago killed it off, and the teams went to the third tied at 3-3. But Verhaeghe cashed in on Florida's next power play, getting what became the game winner just 2:44 into the final period.
Blackhawks: Host Tampa Bay on Thursday.
Panthers: Visit San Jose on Tuesday.
AP NHL: https://apnews.com/hub/nhl