The three-goal second-period flurry triggered the win, which ended a 3-0 homestand.
"We knew they played last night," said Simmonds, referring to the Islanders' hard-fought 4-3 loss in Pittsburgh on Friday. "We thought we played a good first period, but we knew we had more. I think we came in and turned it up in the first five minutes" of the second period.
"Unfortunately, we didn't follow it up in the next 15."
The Islanders, who seemed reenergized after a second-period goalie switch, scored late in the second period - John Tavares ended the Flyers' 345:39 streak without allowing an even-strength tally - and early in the third to get within 3-2 before Schenn netted his fourth goal in the last six games.
Eric Boulton, a fourth-line Islanders winger, cut the deficit to 3-2 with 17:21 left in the third period. The Islanders controlled the next three minutes, causing coach Craig Berube to call his timeout.
Earlier in the year, "we would have [panicked]" when an opponent rallied to within 3-2, Jake Voracek said. "But right now, we're very calm on the bench . . . and we got the fourth goal, which was huge."
The Flyers steadied themselves, and good forechecking by Voracek and Giroux set up Schenn's goal, a rebound from the slot with 7:39 remaining to make it 4-2.
Read, who has four goals in his last two games, added his second tally of the night by scoring an empty-netter with 48.6 seconds left.
Goalie Steve Mason (7-7-2) made 36 saves.
Read scored on a rebound to give the Flyers a 1-0 lead after 46 seconds of the second period. Fifty-nine seconds later, Simmonds took a feed from Vinny Lecavalier and beat goalie Kevin Poulin with a right-circle shot that went high and to the short side.
Giroux then converted a no-look pass from Simmonds to make it 3-0 with a left-circle blast, sending Poulin to the bench and bringing in Anders Nilsson for his first NHL game this season. It marked the sixth straight game the Flyers had scored a power-play goal.
"We've been through a little bit, obviously, this season," Simmonds said. "... Since the coaching change, we've stuck to our structure and we're gaining confidence game in and game out, and we're just getting better every game."
"The players have listened and bought in and worked hard," said Berube, who became the coach when the team was 0-3.
The Flyers are 32-4-2 against the Islanders since 2007-08.