The number of games the team won is W.
The team won one more game than it lost, so W = L + 1.
The total number of games played is 23, so W + L = 23.
We can substitute the first equation into the second equation:
(L + 1) + L = 23
2L + 1 = 23
2L = 22
L = 11
The team lost 11 games.
The team won W = L + 1 = 11 + 1 = 12 games.
Therefore, the team won $\boxed{12}$ games.