The 21-year-old will play for the lower league club until January, while the move could be extended if all parties are happy at that stage.
The England youth international has been a regular for the Arsenal under-23s, but now gets a taste of regular first team football.
Bola told the Rochdale website: "I'm pleased to be here and can't wait to get going.
"There has been talks for a couple of weeks and I'm just happy it got over the line and I can start the hard work.
"I've spoken a lot with Brian Barry-Murphy [manager] and the coaching staff. It's really good to hear his ideas and I think we can have a successful season this year."