John, Carl and Tim worked together. John didn't like Tim at all. So, he started pitting Carl against Tim. John told Carl that Tim was a very bad person, he didn't like Carl as well and Tim used to backbite about Carl. So Carl thought that whatever the John was saying was true and got into a fight with Tim."
You could use "pit" in a more directly confrontational situation.
John is a manager and the other two are in sales. John pitted Carl against Tim when he told them that whichever of the two made the most sales that month would receive a $100 bonus.
The two are "fighting" over a specific thing. The "fight" for customers was directly engineered by John, whereas in your sentence it is an indirect result of John's having turned them against each other.