Win over is the correct phrasal verb though win round is some times used but I would not prefer the preposition round. Win over being a transitive phrasal verb, is separable and as such the object can be placed between the verb and the preposition or after the preposition if the object is not a pronoun
So you can saY:
We'll be working hard over the next ten days to win over the undecided voters.
I hope I can win them all over to our side.
I won over the mayor to our side.