I am certainly no expert in sports talk, but I took it to mean something like this: the team is playing in a series of match-ups of, say, three games against each opponent. They won the first game of the first series. They won the first game of the second series. They won the first game of the third series. They won three consecutive "Game #1"s.
Now, whether it should be "game ones" or "games one" - technically, following the pattern of "attorneys general" and "brothers-in-law" I suppose it should really be "games one" -- but there are two problems with that. First, few people speak correctly enough to know that rule, and, second, "games one" is too easily confused with "games won." I think the best way to deal with it would be to say "the team won consecutive First Games."
Somewhere, perhaps in The Onion (a satirical news-magazine parody) there was an item about William Safire, who writes a language column for The New York Times, going into Burger King and ordering "Two Whoppers, Junior." (Instead of "two Whopper Juniors" as most people would say.) I imagine he would say "games one." But you can't expect that level of linguistic sophistication from a sports announcer.