The Giants moved the ball sharply, at times impressively. Then they settled for a field goal.

Repeat.

On a solid night for Daniel Jones — 22 for 32 for 249 yards passing, and 95 yards rushing — the Giants couldn’t convert when it really mattered on Thursday night in Washington. Five times, their offense headed back to the sidelines and turned the game over to kicker Graham Gano. Five times, he made the field-goal attempt presented to him. But the points left on the board were costly in the Giants’ 30-29 loss to the Washington Football Team.

The cruelest twist? Washington won the game on a 43-yard field goal of its own, a penalty-aided second attempt by Dustin Hopkins with no time left.

Both the Giants and the Football Team were eager for a win after losing in Week 1. Last season it only took a 7-9 record for Washington to win the woeful N.F.C. East. That gave even the 6-10 Giants plenty of hope going into this season. A 0-2 start — for the fifth straight year — is not what they had in mind.

Jones’s good passing performance was matched by Washington’s Taylor Heinicke, who took the reins after the Football Team player Ryan Fitzpatrick was injured in Week 1. It was just Heinicke’s second career start at age 28, but he was 34 for 46 for 336 yards and crucially led his team to three touchdowns to the Giants’ two.

The game seesawed back and forth. The Team took a 27-26 lead with 4 minutes 33 seconds left on an acrobatic, spinning touchdown catch by Ricky Seals-Jones.

After Washington got the ball back, Heinicke made an ill-advised pass at his 22 that was intercepted by James Bradberry with 2:22 left. The Giants suddenly had a chance to steal the game.

Once again, though, they settled for a Gano field goal and a 29-27 lead.

Needing only a field goal to steal the win back, Washington executed a classic 11-play, 50-yard two-minute drill. Hopkins missed a 48-yarder at the end of it, but an offside penalty against Dexter Lawrence gave him a second shot from five yards closer, at the 43. With no time on the clock, he converted it and the Team (1-1) was the winner.

The Giants also got bad news in the game after offensive lineman Nick Gates, a co-captain, went down with a gruesome leg injury in the first quarter. He was carted off the field, and the team later said he had fractured the leg, very likely ending his season.

Much ado was made about receiver Kenny Golladay giving Jones an earful on the sidelines late in the game. Golladay did not meet the press, but Jones said: “I think he was frustrated at the situation. I don’t think it was to me or anyone in particular.”

It was the kind of frustration felt by many Giants fans, still awaiting their team’s first playoff victory since a Super Bowl win in 2012.