Simmonds assisted on Brayden Schenn's goal 2:06 into the game and then scored twice in a 6:54 span to give the Flyers a three-goal lead 14 minutes in.
Surging Philadelphia, which beat its cross-state rival 4-0 at home Saturday, improved to 5-1-1 in its past seven.
Defensemen Brooks Orpik and Matt Niskanen and recent call-up Jayson Megna scored for the injury-depleted Penguins, who lost two straight in regulation for the first time since dropping three in a row from Nov. 6-13.
During the final seconds, the Penguins' Sidney Crosby hit the post with a backhand off a pass from Evgeni Malkin while skating down the right wing.
Pittsburgh still holds a commanding lead in the Metropolitan Division, but remained three points behind the Boston Bruins in the race for the Eastern Conference's top seed.
Matt Read had his third goal of the weekend for Philadelphia, which entered Sunday holding down the eighth and final spot in the conference standings. The Flyers are 8-1-1 in their past 10 regular-season games in Pittsburgh.
Making starts on consecutive days for only the second time this season, Marc-Andre Fleury made a lunging glove save on the first shot he faced Sunday, from Simmonds.
But he couldn't control the puck in his glove and it fluttered to Schenn in the slot, who slid it in past a diving Penguins defenseman Robert Bortuzzo for his 17th of the season.
Claude Giroux assisted on Simmonds' 20th and 21st goals, the first of which was with 13:13 left in the first when he deflected a Kimmo Timonen shot from the point. Simmonds made it 3-0 when he corralled a rebound while uncovered and beat Fleury to end his afternoon after just 13:41.
Under siege by a Philadelphia attack that Pittsburgh had issues dealing with, Fleury allowed seven goals on 55 shots in less than four periods covering 24 hours against the Flyers.
Rookie Jeff Zatkoff replaced him.
Without their top two goal-scoring wingers - Chris Kunitz (lower body) and James Neal (concussion) - for the second consecutive game, the Penguins were held to fewer than four goals for the eighth consecutive game. That hadn't happened to them since early in the 2011-12 season.
A pair of defensemen pulled Pittsburgh to within a goal with slap shots from the point that found their way through traffic and past Flyers goalie Steve Mason.
Orpik was credited with his second of the season with 2:27 left in the first, and Niskanen tallied his 10th at 5:50 of the second.
But Read had a short-handed goal for the second straight day, finishing a 2-on-1 break himself with a wrist shot for his 19th of the season.
Megna tapped in a nice pass from Brian Gibbons to cut the Penguins' deficit to 4-3 with 3:41 left in the second, but Pittsburgh failed to score on 11 third-period shots.
Pittsburgh has the NHL's No. 1 power play and No. 2 penalty kill but lost the special teams battle 5-0 over the weekend against the Flyers.
Notes: Penguins D Simon Despres was a healthy scratch. ... Including playoffs, the Flyers are 10-2-1 all-time at Consol Energy Center (8-1-1 in the regular season). ... Giroux has five assists in the past three games. ... The Penguins fell to 18-2 in March home games over the past three seasons. They were 10-0 at home last March. ... Crosby, who had 74 points over his first 45 career games against Philadelphia, was scoreless in the home-and-home series.