Bills safety Jairus Byrd corrals Eagles quarterback Michael Vick as Buffalo improved to 4-1 and the Eagles dropped to 1-4. (DOUG BENZ/Reuters)

The Philadelphia Eagles at least used a different script. This time, they were the team coming back in the fourth quarter. But their rally fell short and their season continued to unravel as quarterback Michael Vick threw four interceptions and the Eagles lost to the Buffalo Bills, 31-24, on Sunday at Ralph Wilson Stadium.

The Eagles lost their fourth straight game to drop to 1-4. They are in last place in the NFC East, and their hopes for the sort of turnaround that they would need to fulfill their lofty preseason expectations are dwindling rapidly. They play the division-leading Washington Redskins next weekend at FedEx Field.

“I’m not looking at [the] playoffs or anything else,” Eagles Coach Andy Reid said. “I’m looking at one game at a time. We’ll fix the problems here and then we’ll get on with Washington. . . . We have no excuses. We’ve got to get better.”

Vick ran for 90 yards and became the NFL’s career rushing leader among quarterbacks. He also threw for 315 yards and had two touchdown passes, one to wide receiver Jeremy Maclin in the second quarter and one to wideout DeSean Jackson in the opening seconds of the fourth quarter. Tailback LeSean McCoy added a third-quarter touchdown run for the Eagles.

But Vick threw three interceptions in the first half while the Bills, who upped their record to 4-1, crafted a 21-7 lead. Bills tailback Fred Jackson ran for a touchdown. Quarterback Ryan Fitzpatrick threw a touchdown pass to wide receiver David Nelson, and linebacker Nick Barnett scored a touchdown on an interception return.

Brad Smith added a third-quarter touchdown run on a direct snap out of the Wildcat formation, increasing the Bills’ advantage to 28-7.

The Eagles, who had squandered fourth-quarter leads in their previous three games, scored 10 straight points after trailing, 31-14, entering the fourth quarter Sunday.

But they failed to give the NFL what would have been its fifth comeback victory this season by a team that trailed by 20 or more points. The Eagles drove to the Buffalo 25-yard line and were seeking a tying touchdown when Barnett got his second interception of the day on a pass by Vick that wide receiver Jason Avant had knocked from his hands by cornerback Drayton Florence.

“We preach turnovers throughout the week and we were able to get some today,” Florence said. “We made plays when we needed to make them at the end of the game.”

The Bills ran out the clock after converting on fourth and inches near midfield when Eagles defensive end Juqua Parker jumped offside.

“There isn’t one person blaming Juqua at all,” Eagles defensive end Jason Babin said. “It isn’t one play losing the game. We’ve got to get back to work. We’ve got get back to the basics and just prepare.”

Vick threw interceptions on each of the Eagles’ first two possessions. In between, the Bills scored on a five-yard touchdown run by Jackson.

The Eagles got even early in the second quarter on Vick’s two-yard touchdown pass to Maclin. That came after the Eagles retained possession on an instant-replay reversal of what originally was ruled a lost fumble by Vick. Instead, Vick was called down by contact before the fumble.

But the Bills took control from there. Jackson turned a screen pass into a 49-yard gain to set up Fitzpatrick’s six-yard touchdown pass to Nelson. On the Eagles’ next offensive play, Vick was hit by linebacker Kelvin Sheppard as he delivered a pass. The throw went directly to Barnett, who made the interception and went 31 yards for a touchdown. That gave the Bills touchdowns on interception returns in three straight games, the first time they’ve done that since 1960.

“I should have just taken the sack,” Vick said. “I was trying to force it, not trying to take the sack. You’ve got to be disciplined in the moment. Sometimes it’s hard but I think that was a game-changer. You go from 14-7 to 21-7. . . . Those things can’t happen.”

Vick said that he and other Eagles players are making mistakes while trying too hard to get things turned around.

“We know what we’re capable of,” Vick said. “We know what we can do. We move the ball up and down the field all game. . . . Guys are just trying to make the play.”

Said Reid: “I think we need to eliminate turnovers . . . and penalties in crucial situations. I think it’s pretty obvious.”

The Eagles had a chance to draw closer in the final moments of the first half. But they opted to try one more play on third and 10 from the Buffalo 26-yard line rather than attempting a field goal, and time expired in the half as Vick threw an incompletion.

“I thought we could get one more play and have some time,” Reid said.

The Bills recovered an onside kick by the Eagles to open the second half. But the Eagles got a reprieve when the officials ruled that the kick had been taken before the ball was declared ready for play.

No matter. The Eagles kicked off into the end zone for a touchback, but the Bills went on a drive capped by Smith’s five-yard touchdown run. That made it a 28-7 game, but the Eagles used Vick’s 53-yard scramble to set up McCoy’s 10-yard touchdown run.