It is necessary. You can't say this without the 'against'.
Arsenal (note spelling) did not win Chelsea. This makes no sense. You can win a game. You can win a championship. You can win a cup. But you can't win a rival team. You BEAT the other team. Arsenal beat (or even defeated) Chelsea 3-0.
If you say 'Arsenal won against Chelsea 3 - 0', what you really mean is 'Arsenal won [the match] against Chelsea 3 - 0'.