From then until the founding of MLS in 1996, it's been won by the Greek American Athletic Club in 1967, 1968, 1969, 1985 and 1994; Elizabeth Soccer Club in 1970 and 1972; New York Hota in 1971; Brooklyn Dodgers Soccer Club in 1979; the New York Pancyprian-Freedoms in 1980, 1982 and 1983; New York AO Krete in 1984; and the Brooklyn Italians in 1991.
Since MLS was founded, the only U.S. Open Cup Finalists not to be MLS teams were the Rochester Rhinos in 1996 and 1999 (winning the latter), and the Charleston Battery in 2008.
I am not including the Major Indoor Soccer League. Given the smaller field, and needing to get the ball over the hockey boards at the arenas for it to go out of bounds, it was a very different game.
10. 1967-68 New York Generals. The 1st "major league" soccer team in New York, they played at the old Yankee Stadium, finished 3rd both seasons, and didn't make the Playoffs. They're just here to fill out the ranks.
9. 1996-2000 New York/New Jersey MetroStars. In their 1st 5 seasons, they made the Playoffs 3 times, and 3 times reached the Semifinal of the U.S. Open Cup.
8. 2001-05 New York/New Jersey MetroStars. In their 2nd 5 seasons, they made the Playoffs all 5 times, but lost in the Conference Semifinals all 5 times. In 2003, they reached the U.S. Open Cup Final, losing to the Chicago Fire. In 2006, they were bought by Red Bull, and the name was changed, although many fans still call them "Metro" and come to games wearing old MetroStars jerseys.
7. 2015-19 New York City FC. They have now played 5 seasons, with considerably more hype than the Red Bulls, if not with a stadium they can call their own. In the regular season, they finished 2nd in 2016 and '17, and 1st in '19.
But, after missing the Playoffs in their 1st season, they have made the Conference Semifinals 4 straight seasons, but never the Conference Finals. The closest they've come to the U.S. Open Cup is this year's Quarterfinal. Next year will be the 1st CONCACAF Champions League for which they've qualified.
6. 2006-08 New York Red Bulls. 3 seasons, 3 Playoff berths, the Quarterfinal of the 2006 U.S. Open Cup, and the 2008 MLS Cup Final, losing to the Columbus Crew. This remains the closest that the MetroStars/Red Bulls have ever come to winning a league title. Then came 2009, and the worst season in franchise history. Then came the move from the Meadowlands to Harrison, and a new era.
5. 2016-18 New York Red Bulls. Won the 2018 Supporters' Shield for MLS' best regular-season record. Got to the Conference Semifinals in 2016 and '17 and the Conference Finals in '18, and to the Final of the 2017 U.S. Open Cup, losing to Sporting Kansas City. Got to the Quarterfinals of the CONCACAF Champions League in '16 and '17, and the Semifinals in '18. But it all fell apart last season, ending this run.
4. 2010-15 New York Red Bulls. 6 seasons, the beginning of the Red Bull Arena era, and (through 2014) the Thierry Henry era. Won the Supporters' Shield in 2013 and 2015. Got at least to the Conference Semifinals every year, and to the Conference Finals in 2014 and 2015. Got to the Quarterfinals of the U.S. Oen Cup in 2011 and 2015. But, for varying reasons, didn't really get close to a title.
3. 1971-73 New York Cosmos. In their 1st 3 seasons in the old NASL, they finished 2nd, 1st and 2nd in their division, and reached the Semifinals of the Playoffs all 3 times. In 1972, they won the title.
2. 1981-84 New York Cosmos. 4 seasons, 3 division titles, the 1981 NASL runners-up and the 1982 Champions. This team was a bit different from the 1976-80 "dynasty." Chinaglia and Carlos Alberto were still there, but now, they had Rick Davis, Vladislav Bogićević, Dutch stars Wim Rijsbergen and Johan Neeskens, Steve Wegerle, Steve Hunt, Andranik Eskandarian, Seninho, and Hubert Birkenmeier in goal.
1. 1976-80 New York Cosmos.5 Playoff berths, and the 1977, 1978 and 1980 NASL Championships. Still the greatest soccer team in North American history, and if you think any MLS or Mexican team was better, then you must be smoking something Mexican. This team had Pelé, Carlos Alberto, Franz Beckenbauer, Giorgio Chinaglia, Jomo Sono, Werner Roth, Bobby Smith and Shep Messing.
It's hard to measure the Cosmos against the great teams of European and South American "football," since the competition they were facing wasn't exactly top-flight. But they did win 5 league championships, and the only other Tri-State Area teams to have done that are the Yankees and both the baseball and the football Giants.
Seriously, here's the count: Yankees 27 (counting World Series wins, rather than Pennants, as "league championships"), Giants (football) 8, Giants (baseball) 7, Cosmos 5, Rangers 4, Islanders 4, Devils 3, Mets 2, Knicks 2, Jets 1, Dodgers 1, Nets 0 (although 2 in the ABA), Liberty 0 (although 4 Conference Championships), Red Bulls 0 (although 3 Supporters' Shields), NYCFC 0.