Win round
Definition 1 of 1
Pro Tip 1/3Persuasion Process
The phrase often implies a gradual process of convincing someone over time. 
He finally managed to win round the committee after several meetings.
Pro Tip 2/3Positive Change
'Win round' generally refers to changing someone's mind in a positive way, leading to agreement. 
Despite her initial doubts, I won her round with the detailed project plan.
Pro Tip 3/3Emotional Effort
Using 'win round' can suggest emotional effort was involved in persuading someone. 
It wasn't easy, but he won his friend round with heartfelt pleas.