The world number one almost missed the first hole on Sunday after misreading the original tee time of 11.25am, when he was due to face the USA's Keegan Bradley.
"He was receiving a lot of phone calls en route," Rollins told BBC Radio Five Live. "We had minimal conversation but he was a gentleman.
"He was nervous. If I was in his shoes I'd have been as well. I had gone to the hotel to check in with our officers. I realised that one player had not come down from their room to get transported by the drivers to the course. He rode in the front passenger's seat with me. We whisked him away up to the course.
"I'm getting ribbed at work for this but I am very proud of our force and our community, we did the right thing."
McIlroy won his match against Bradley to help Europe retain the Ryder Cup in Chicago.