In challenging conditions, the American touring team took the lead with their first shot on target – thanks to a helping hand from the strong wind.
Rovers responded well and eventually equalised just before the break through Amy Bland.
The hosts continued where they left off after the interval, putting Ohio under plenty of pressure.
Rovers eventually forced an own goal to go 2-1 ahead, before Katie Anderton added a third. Hannah Williamson – the smallest player on the pitch – completed the scoring with a header.
New manager John Prince said: “It was a very strong performance. We got the ball down and dominated possession.
“Ohio were the ideal opposition, as they were a young, fit side, who worked very hard.”