I played Tomasz at tennis. I won with him two games, and he won with me three games.
This one is another example of a calque, or to put it another way, a word or phrase that has been borrowed from English and literally translated word-for-word.
TEE has a few examples of calques, but for this example, the correct phrases should be: I played Tomasz at tennis (or played tennis with Tomasz, which is more natural). I beat him at two games (or I won two games) and he beat me with three (games) (Or he won three).
In this context, to win means to defeat someone or something such as a competition by doing better than the opponent(s): Leicester City have won the Premier League! I won the Chess tournament. Although to beat has pretty much the same meaning, the verb requires an object. So you have to beat someone or something: Leicester City beat Manchester City (the object) by three goals to one. There are several people (the objects) I had to beat to win the Chess tournament. Those bigger players (the objects) will be difficult to beat.