You can certainly have multivariate regression models where the response is not a single value but a vector of possibly dependent values.
You can also have more than one input (multiple regression), so "Team 1 performance" and "Team 2 performance" (or some joint transformation of them, such as their sum and difference, perhaps) could both be predictors in such a model.
So at first glance, multivariate multiple regression would seem to be a good starting point.
The next thing to consider might be whether the response would be better modelled as something other than multivariate normal. You might look into GLMs, for example.
In addition, when looking at pairwise contests like this, it can be difficult to get any absolute measure of individual team performance since they depend heavily on what the other team does - this might lead toward something like Bradley-Terry type models.