The Giants followed a new script on Sunday but still couldn't stage their first successful ending of the 2020 season.

After scoring an NFL-low 47 points through the season's first four weeks, the Giants erupted for 34 points, 18 more than their previous season high, in AT&T Stadium. Their 20 first-half points were more than they had scored in any full game. The team that hadn't held a lead since opening night jumped out in front by 14 points.

Despite all that, Greg Zuerlein's third field goal of the game and second in the final two minutes, a 34-yarder as time expired, left the Giants with a 37-34 loss to the Dallas Cowboys and an 0-5 record.

"This one hurts, because we feel like we could have won that one," safety Logan Ryan said. "We didn't play well enough to win."

"I think it's been a tough start," quarterback Daniel Jones said. "To say we aren't disappointed would be a lie. We certainly are, it's been a tough start. I don't think anyone on this team is discouraged. We're not going to allow ourselves to be discouraged and let this start affect the rest of this season."

The Giants hurt themselves with a turnover that turned into a Dallas touchdown and penalties that took away two of their own scores. Jones and the Giants had a chance to win the game but punted after a final offensive possession that lasted only 58 seconds. The defense failed to stop Dallas at the end of the first half and the end of the game, the latter even after quarterback Dak Prescott was carted off with a compound fracture of his right ankle in the third quarter. The 2-3 Cowboys scored their final 13 points with Andy Dalton at quarterback.

"I heard Logan say it best, it's at this point in time that there's enough of this, 'We're so close, we're getting there,'" linebacker Blake Martinez said. "It's time to win, this is a result business, we need to go out there and win, get the job done, and it's time for that now."

Each team scored a touchdown on an unusual play. For the Giants, it was Evan Engram's three-yard run, the first touchdown run by a Giants tight end since Aaron Pierce did it on Sept. 29, 1996. Not to be outdone, Prescott became the first opposing quarterback in the Super Bowl era with a touchdown reception against the Giants when he caught an 11-yard pass from wide receiver Cedrick Wilson in the second quarter.

In addition, both sides scored defensive touchdowns, the Giants on linebacker Kyler Fackrell's 46-yard interception return and Dallas on a 29-yard return of a Jones fumble by Anthony Brown.

The Giants, who also scored on Devonta Freeman's four-yard touchdown run, got field goals of 55, 50, 54 and 28 yards by Graham Gano, who became the 11th player in NFL history and first for the Giants to kick three field goals of 50 or more yards in a game.

Dallas countered with Ezekiel Elliott touchdown runs of one and 12 yards and the three field goals by Zuerlein, who became the first kicker to score the deciding points against the Giants as time expired since Tampa Bay's Nick Folk on Oct. 1, 2017.

But while they made plenty of plays, it was those they missed that haunted the Giants.

With the score tied 17-17 late in the second quarter, Engram never left the field on fourth down and lined up wide to the right as the Giants prepared for what appeared to be a field goal attempt. The Cowboys never noticed him until holder Riley Dixon threw a 27-yard pass to Engram for an apparent touchdown. But the play was nullified by an illegal shift penalty. Gano kicked his second field goal, but the four lost points proved to be decisive.

"I was mad at our execution that took points off the board," coach Joe Judge said. "That was it, point blank. You prepare for something like that, you call it at a certain time, it came up, you want to see it work. We've got to clean that up."

The Giants were poised to take their three-point lead into the halftime locker room before Dallas drove 75 yards in six plays, the last Prescott's touchdown reception.

Trailing 24-20, the Giants appeared to take back the lead when Jones connected with Darius Slayton for a 31-yard touchdown pass on the first possession of the third quarter. But that score was taken off the board because of an offensive pass interference penalty by fellow wideout Damion Ratley.

"The play wasn't designed to pick anybody," Slayton said. "It was something that happened incidentally. I wasn't expecting that, but it happens."

But the Giants continued to fight and owned a 34-31 lead after Freeman's touchdown and Jones' two-point conversion pass to Andrew Thomas – yes, the rookie left tackle, who became the first offensive lineman in league history to score on a two-pointer.

The Cowboys tied the game on Zuerlein's 40-yard field goal with 1:56 remaining. That gave the Giants an opportunity to hold the ball and give Gano a chance to win the game in the final moments. But after picking up one first down, Jones' pass to Dion Lewis was incomplete, forcing them to punt.

"We were trying to be aggressive to get down the field," Judge said. "We knew the situation, wanted to get a first down, give ourselves a chance to get in field goal range and win the game. There are different ways of managing it and you match different ways in different games. We always talk through those decisions, we're on the headsets together, making sure we're on the same page."

"Tough situation, ball's in our hands and we've got to find a way to win the game," Jones said.

Dallas did. Starting at their own 12 with just 52 seconds left and Dalton at quarterback, the Cowboys executed consecutive completions of 15 yards to Amari Cooper and 19 and 38 yards to Michael Gallup. Both completions to Gallup were so close they required confirmation via replay review, the last with only three seconds left. That was enough time for Zuerlein to kick the game-winner.

"At the end of the day, it's about executing your 1/11," Martinez said about the defensive shortcomings at the end of either half. "I think every single time we get in those situations, there's always just one play here, one guy here, one person here, it's kind of a domino effect where you think you're good, and all of a sudden, someone else thinks they're good, and it's been kind of a splinter in our foot."

The Giants will try again to get that elusive first victory next week at home against the Washington Football Team.

"We have a lot of football ahead of us, and I think everyone on this team feels like we've been in position to win games, we've improved as a team, and we're capable of winning each of these games," Jones said. "So, in terms of our confidence as a team, and what we believe we're capable of doing, I think that's going to be the important thing in us finishing strong and winning these games down the stretch. So, that's where we are as a team, we're disappointed, we want to win every game, but we're not going to let it affect our preparation and how we approach these next 11 games."