Tampa Bay had a good chance in the sixth inning when Molina sent a drive to leftfield with a man on second base, but Steve Lombardozzi raced in to make a diving catch to end the inning.
"I took one or two steps back on it at first, misjudged it, but I sprinted in and was able to catch it," Lombardozzi said.
"I was just hauling my butt in and was able to come up with the catch."
McIlroy carded an eagle and five birdies to offset two bogeys, and narrowly missed out on a second eagle when he lipped out a putt of less than four feet on the par-five 16th hole.
The Briton found the groove on his drives and fired away at the pins, knocking several approach shots close.
"I saw a lot of positive things out there," McIlroy said. "I felt like I drove the ball great on the last few holes and hit some great approach shots."
A birdie at the second hole followed by an eagle at the par-five third helped put McIlroy in the right frame of mind.
"Three-under through three holes gave me a little bit of a boost today," he said.
Earlier, Holmes had tore up the front nine to sit pretty at the top of the leaderboard after the morning section of play.
The 30-year-old Kentuckian, who has returned well after undergoing brain surgery last September, birdied the second and third holes and ran off four straight birdies from the sixth.
