Increasing the temperature would also increase the rate of the exothermic reaction, however what drives the molecules present in the system to choose the endorthermic pathway?
Think about it this way:
There are two people. One guy likes chocolates and one guy hates them. They decide to transfer the chocolates. If you start throwing chocolates randomly to both of them, in which side will the reaction go? The guy who hates chocolates would keep giving it to the guy who likes them.
In a similar way, if you provide heat to a reaction, it would move in the direction of the where the heat is absorbed, i.e the endothermic pathway.