Rockies reliever Adam Ottavino (2-0) struck out five Nationals in the 10th and 11th innings, including Danny Espinosa, Bryce Harper and Ryan Zimmerman in order.
"That was a tough one," said Nationals manager Davey Johnson. "I tip my hat to the guys, battled back from a deficit several times."
Harper broke a three-inning scoreless drought with a game-tying homer in the top of the ninth off Rafael Betancourt. He felt it was one of the first good pitches he saw all series.
"They've been pitching me out, out, out, up, out, out," said Harper, who finished 2-for-6 with two RBIs. "They didn't throw me one strike all series. They got some questionable calls from the umps, I thought, but whatever, I don't care. I got a pitch on the plate, drove it, and it's a 10-10 ballgame."
In the bottom of the ninth, Stammen induced Scutaro into grounding into inning-ending double play after giving up a single. Stammen yielded a one-out single in the 10th but got two consecutive ground balls to end the inning.
Washington faced an uphill climb from the get-go -- having to cut into deficits of 7-0, 9-5, and 10-9.
"We're never going to stop," Harper said. "We have more heart than anyone."
Starting pitcher Edwin Jackson gave up eight runs on 10 hits in three innings. It took Jackson 31 pitches to get out of the first inning, in which he was bullied for five runs, swelling his career ERA at Coors Field to 17.40.
"Both teams have to pitch in the same park, it's not an excuse," Jackson said."You have to go out and get the job done, and I didn't do a good job of that today."
The Nationals (43-31) are left to ponder what happened to the team that marched into Denver with a reputation as a strong-pitching, light-hitting team and left town with an Atlanta-bound flight after having scored 33 runs and giving up 21 in the course of three games. For most of the series, they looked like the Rockies.
"It's Colorado," Ian Desmond said flatly.
Desmond was one of many offensive players to enjoy the benefits of the thin air. He tied the game at 9 in the fifth inning with a two-run homer to left field, and then doubled to lead off the 10th inning. Desmond's 37 extra-base hits this year -- he had eight here in Colorado -- lead all Major League shortstops.
"We're confident from the first pitch," Desmond said. "We know we've got a good ballclub, we knew we'd battle back. We gave it a run."
Jackson gave up three consecutive singles to begin the game, then walked Todd Helton. Chris Nelson dinged him for another single to score Dexter Fowler, then Tyler Colvin followed with a three-run homer.
"I did a terrible job of keeping the ball down in the zone and getting ahead," Jackson said. "We battled back, but I feel like that loss is on me. When the bullpen has to come on in the fourth inning, that's never good. I didn't get the deal closed."
Jackson gave up two more runs in the bottom of the second.
Zimmerman poked a two-run single through the middle of the infield in the third inning to get the Nationals on the board against Rockies starter Josh Outman. Two batters later, Michael Morse followed with a towering home run over the center-field wall to score three more and chase Outman from the game.
"You go out and play as hard as you can, we did that today," Desmond said. "We just didn't finish the job."
The Rockies carried a 10-9 lead into the top of the ninth before Harper hit his eighth homer of the season into the right-field bleachers.
"We played a good series, we didn't get that final win, but they played it tough and they swung the bat well," Harper said. "We battled, they battled. The ball rolled their way, there's nothing we can do."