With their win, the Cosmos still have a shot at a berth in the Concacaf Champions League, the top club competition in North and Central America. There are two primary paths to that competition: through M.L.S. play, either by earning the best regular-season record or by winning M.L.S. Cup; or by winning the Open Cup, a tournament open to clubs at all levels of soccer in the United States.

No team outside M.L.S. has captured the Open Cup title since the Rochester Rhinos did in 1999. Then again, there has not been a second-tier team quite like the high-spending Cosmos, who joined the N.A.S.L. in 2013 and feature Raúl and Marcos Senna of Spain.

The Cosmos, whose oft-repeated goal is “soccer at the highest level,” have only one path to the Champions League, and that is by winning the Open Cup.

N.Y.C.F.C. was competing in its first-ever match in the tournament, which M.L.S. teams joined in the fourth round. Nine of its 11 starters could be considered regulars, with only the star forward David Villa and goalkeeper Josh Saunders held out of the lineup.

“Do I think they could compete in M.L.S.? Yes, I do, but the fact is that they play in the N.A.S.L., which is, for lack of a better phrase, the ‘second division,’ ” Petke said before the game. “So, of course, every time they have an opportunity to play M.L.S. teams, they are going to use it to prove themselves.”

Cosmos Coach Giovanni Savarese looks at the game in more of a big picture.

“I think more than the best team in New York,” Savarese said before the game. “That’s more for the fans. We can demonstrate that we can compete and that we are difficult to beat, a team that can play with anyone. We very seriously want to demonstrate in New York that we’re not behind anyone, that we’re a very tough team, a very strong team.”

For his part, N.Y.C.F.C. Coach Jason Kreis used no alibis in describing his team’s loss as anything but legitimate.

“It’s disappointing for sure,” Kreis said after the game. “It’s probably most disappointing because you go through 120 minutes. Probably look back at it and think, if we’re going to lose, probably best if we’d done it in the first 90 minutes.”