The game was immediately stopped as medical staff rushed to his aid when he suddenly slumped to the bench after completing a shift.
Peverley was treated in a hallway then rushed to a nearby hospital for treatment to what a team doctor described as a cardiac event but NHL officials said he was conscious and recovering.
"Dallas player Rich Peverley is doing well and is in stable condition. He has been transported to the hospital," the NHL said.
"As a result of the emotional state of the players on both teams caused by the medical emergency, the game is being postponed.
"We apologise for any inconvenience and we thank the fans."
Dallas coach Lindy Ruff said it was a frightening experience for the whole team.
"I was scared," Ruff said. "My first emotion was we need somebody here real quick.
"When he dropped, it was red alert, don't worry about the game, don't worry about anything else; just turn around and scream for a doctor and that's all.
"It was just let's get him the help he needs, and they came and got him the help. For me, it was something I don't want to witness again."
An eight-year veteran in the NHL, the 31-year-old Peverley won a Stanley Cup with the Boston Bruins in 2011 but was traded to the Stars last year.
The Canadian missed a handful of games to undergo treatment for an irregular heartbeat, including a game last week.
As soon as he collapsed on Monday, his team-mates frantically banged their sticks on the boards to notify officials that was something was wrong.
He was carried from the bench into a hallway where doctors gave him oxygen and used a defibrillator.
"As soon as we treated him, he regained consciousness," doctor Gil Salazar said.
"He was alert and awake talking to us after the event and quickly transported to the hospital.
"At this point, I was able to talk to him in the back of the ambulance, and he was able to tell me where he was and he actually wanted to get back in the game."
Players from both teams remained on the ice for about 15 minutes before the game was called off then retreated to the locker room. The NHL said they would decide at a later date when they would finish the game.
"There's nobody in there that wants to play hockey right now, and I think everybody understands that when you've witnessed what they had to witness and that's their teammate," Ruff said.
"And that's the right place to be. That's the right emotion to have. They're not doing very good, and I wouldn't expect them to be."
- Sports & Recreation