With the forward following the Stars game against the St. Louis Blues from a hospital bed back in Dallas, Benn put his team over the top.
Stars coach Lindy Ruff said Peverley, who is sidelined because of a heart issue that caused him to collapse on the bench on Monday, was well aware his teammates beat the Blues 3-2 in overtime on Tuesday night.
''I texted him right after the game and he responded,'' Ruff said. ''He was happy. He said 'Keep rolling.' I just said at the end, 'See you tomorrow.'''
Benn scored 3:42 into overtime to end the Blues' five-game winning streak. St. Louis is first overall in the NHL standings.
''Obviously, (Monday) night was a scary situation but today is a new day,'' Benn said. ''We were thinking about Rich back home, but I thought we did a great job getting mentally ready for this game.
''We're still in a big playoff push and we found a way to get two points.''
They might have run into an inspired team.
''You've got to give them credit. They've been through a lot in the last 24 hours,'' Blues forward Steve Ott said. ''It's pretty honorable for them to come out and play like that.''
Colton Sceviour, one of two call-ups by the Stars with Peverley out indefinitely and teammate Alex Chiasson also hospitalized due to emotional distress after his teammate's ''cardiac event,'' scored on a power play in the first period.
Miller robbed the other call-up, Chris Mueller, with a glove save from point-blank range with 2:16 to go in the third. But he couldn't cross the crease in time to corral a shot from the right circle by Benn that gave Dallas its fourth win in five games.
''I kind of knew he got lost up there, but no one was on the back side the way the play developed,'' Miller said. ''I just didn't get my whole body over in time.''
The Stars' home game against Columbus was postponed Monday night after Peverley fell ill. Peverley remained hospitalized in stable condition on Tuesday while undergoing tests.
''I told them, 'You can look for a reason to lose or you can find a way to win,''' Ruff said. ''I said we need to find a way to win and we need to win it for a couple of our teammates.''
Benn has scored in three straight games and has seven points in a five-game streak.
The Blues are 18-0-2 against the Central Division, breaking the franchise record of 19 consecutive games with a point in divisional play set in 1968-69. During the five-game winning streak, they never allowed more than two goals.
After settling for one point, coach Ken Hitchcock saw only negatives.
''I don't think we're playing very smart,'' Hitchcock said. ''We're not putting pucks in deep, we're not getting on the grind, we're turning way too many pucks over.
''Players are going to have to dig in and start playing the right way.''
Polak tied it at 2 at 5:01 of the third period with his first goal in 37 games and fourth overall. He sent a drive through traffic that got past goalie Tim Thomas.
Pietrangelo, who earlier this season became the first Blues defenseman to score 40 points in his first three seasons, snapped a 20-game goal drought in the first period.
Pietrangelo had 11 assists since his previous goal on Jan. 10 at Vancouver.
Antoine Roussel scored on a breakaway earlier in the period to put the Stars ahead, sending the puck slowly trickling between Miller's pads. Roussel got free after David Backes flubbed a cross-ice pass near the blue line in the offensive zone.
Sceviour, among the leading scorers in the AHL for the Texas Stars, scored his fourth goal in eight games this season against a team that had killed 33 of 35 power-play chances the previous 11 games.
NOTES: Blues D Jay Bouwmeester played in his 700th consecutive regular-season game, the longest active streak in major sports. ... The Blues fell to 36-1-5 when scoring first. ... Roussel had one assist the previous four games. ... Polak has four goals, tying his career best for a season.