Woods' only victory in which he didn't have the 54-hole lead here came in 2009, but then, he was alone in second and overtook Sean O'Hair.
"I felt like I left one or two out there coming in and I definitely could have wrapped up a two or three-shot lead for tomorrow which would have been a nice cushion, but it didn't happen", he said.
McIlroy missed the cut last week.
"I started the day just outside the top 10 and wanted to at least give myself a chance going into tomorrow, so it was a great day out there", he said.
England's Rose made six birdies against a lone stumble, a three-putt bogey at 15, to shoot 67 and share fourth on 207 with American Ryan Moore. Charley Hoffman, Rickie Fowler, Talor Gooch and South Korea's Byeong Hun An settled in at 8 under and a tie for sixth.
Woods, coming off a runner-up finish last week at Innisbrook, is an eight-time victor at Bay Hill, most recently in 2013 when he last played the Arnold Palmer Invitational.More news: That Bottled Water You're Drinking May Contain Tiny Particles of Plastic
"I know Phoenix", Woods said, referring to the Waste Management Phoenix Open, "guys were telling me this year, they were yelling and trying to time it". He immediately bounced back though, making eagle at the par-5 12th to get to three under on the round. "I'm going to have to probably get a little bit of help, but my responsibility is to go out and shoot a low one first".
Stenson played bogey-free for a 3-under 69 and goes into the weekend with another shot at winning Bay Hill. Each time he needed two putts to walk off with a birdie.
But an errant tee shot at the par-5 16th led to a bogey and he dropped another shot at the 17th to fall well off the pace at 10 under.
Before play began at Bay Hill, the 14-time majors victor Woods said: "I'm very excited".
When he slipped home a 12-footer to birdie the 18 and ignite more thunder, Woods casually assessed the situation and stayed unfazed by the commotion.
Following opening rounds of 68-72, Woods posted a respectable 3-under 69 in the third round, thanks to six birdies against three bogeys.
Woods also holed from 11 feet for par on the ninth, his final hole, after finding a greenside bunker with his approach and is now vying for favouritism for the Masters with world number one Dustin Johnson.