EAST RUTHERFORD, N.J. (AP) — The Super Bowl champion New York Giants are off to their best start in eight years, and their biggest concern heading into the bye week seems to be whether they can keep two kickers.

The Giants (3-0) are finding ways to win, taking advantage of a pass breakup from veteran Sam Madison, a clutch catch from Amani Toomer and another straight kick from 44-year-old relief kicker John Carney.

All three contributed in overtime on Sunday in a 26-23 victory over the winless Cincinnati Bengals.

Madison, who has been limited following offseason hernia surgery, broke up a third-down play to get the ball back for New York. Toomer later tiptoed the sideline on a 31-yard catch that set up Carney’s winning 22-yard field goal, and the Giants won their seventh straight game dating to the playoffs last season.

‘‘That’s a game that wasn’t pretty at all times,’’ Giants quarterback Eli Manning said. ‘‘We didn’t play our best football — that’s how it goes sometimes. It’s not going to be perfect, we had ups and downs but we kept fighting. When we needed big plays, guys stepped up. We’ve always had that ability, had that about us — it’s good that we haven’t lost that.’’

The 3-0 start is the Giants’ best since they won their first three games en route to a 12-4 season in 2000, a year the team went to the Super Bowl, losing to Baltimore.

What has made this start so impressive is that it’s come after the retirement of seven-time Pro Bowl defensive end Michael Strahan and the loss of Pro Bowler Osi Umenyiora to a season-ending knee injury.

Players have stepped up on both sides of the ball with Carney being the least likely. He was signed just days before the season because Lawrence Tynes injured a knee in training camp.

Tynes is getting close to returning, but it might be hard for coach Tom Coughlin to replace Carney, who has made all nine field goal attempts and done a great job on kickoffs.

‘‘I realize at this point in my career, and this opportunity, I am a relief pitcher. When my job is done I will go back to the bullpen and hope for another opportunity somewhere around the league,’’ said Carney, who has kicked three overtime field goals in his 21-year career.

The play that set up Carney’s fourth field goal Sunday was the long pass from Manning to Toomer down the left sideline on a third-and-10 from the Bengals 38.

It was difficult to tell whether Toomer got both feet in bounds. The Giants hustled to the line of scrimmage and handed the ball to Derrick Ward for a 3-yard run to the Bengals 4, precluding a video review.

‘‘I couldn’t tell,’’ Toomer said when asked if he was in bounds. ‘‘I just got as many feet down as I could.’’

Coughlin said the umpire moved off the ball, so the Giants knew that the replay official had allowed the play to stand. The Bengals wanted referee Jeff Triplette to review the play, but he couldn’t do it in overtime unless asked by the replay official.

‘‘I don’t know if that was a completion or not,’’ said Bengals receiver T.J. Houshmandzadeh, who had 12 catches for 146 yards and a touchdown. ‘‘But, yeah man, you have to review that. It’s too close. I don’t know, I was on the side. I thought it was an incompletion obviously. But yeah, it was close. I thought they would review it, they didn’t. It was a catch. They won.’’

NFL spokesman Greg Aiello said the replay official looked at the play and ruled it legal.

The 0-3 start is the worst for the Bengals since 2003, Marvin Lewis’ first season as coach.

‘‘It’s definitely a tough situation, a situation we never envisioned ourselves being in,’’ said Carson Palmer, who was 27-of-39 for 286 yards. ‘‘We’ve got to keep coming together as a team. It may sound crazy, but we have improved, but not enough.’’

The final five minutes were wild.

Cincinnati took a 20-16 lead on a 17-yard pass from Palmer to Houshmandzadeh. The Giants came right back and went ahead 23-20 on a 4-yard TD toss to tight end Kevin Boss with 1:50 to go.

Palmer drove the Bengals downfield again, hitting Houshmandzadeh with passes of 20, 8, 16 and 9 yards to set up Shayne Graham’s tying 20-yard field goal on the final play of regulation.

Graham also kicked field goals of 22 and 30 yards for the Bengals. Chris Perry, who ran for 74 yards on 20 carries, scored on a 25-yard run.

Brandon Jacobs had a 1-yard touchdown run for New York, and Carney added field goals of 24, 46 and 26 yards.

Until overtime, the Bengals appeared ready to borrow a page from the Giants’ road to the Super Bowl last season. After an 0-2 start, the Giants went on the road and beat Washington to turn around their season — a game that ended with a goal-line stand.

The Bengals, who scored a total of 17 points in losing to Baltimore and Tennessee, had a chance for the goal-line stand on Sunday, but failed to make it.

Cincinnati leads the overall series with the Giants 5-3 with the home team winning every game. … The Giants had six sacks for the second straight week. DT Fred Robbins had two for a second consecutive game. … Former Giants LB Dhani Jones led the Bengals with 12 tackles. MLB Antonio Pierce had 13 for New York. … Cincinnati’s Chad Ocho Cinco had three catches for 29 yards, with 15 coming on a short flea flicker. … Ward had 80 yards rushing on nine carries.