Overnight leader Stenson emerged from the pack after duelling with Bryson DeChambeau on day three at Bay Hill, posting a one-under-par 71.
McIlroy has missed the cut in two of his last four starts on the PGA Tour but remains in contention after a second-round 70 that featured five birdies and three bogeys.
The 14-time major victor was two-over after his front nine, but at least managed to salvage some hope on the back nine with birdies at the 12th and 16th, that saw him finish with a 72.
"I can't really ask for much more".
Rose's playing partner Woods also clawed his way back into the tournament with a 69 and at seven under lurks ominously.
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The 11th-seeded Bonnies (26-7) will play sixth-seeded Florida (20-12) in Dallas on Thursday night in the East Region. If Nova reaches the Final Four, it will set a Division record for wins in a four-year span (Gonzaga also could).
That overall total of 7-under, though, leaves him a handful of strokes behind the current leaders, while fellow contenders include the likes of Henrik Stenson, Rickie Fowler, Justin Rose, Patrick Reed and Rory McIlroy.
At the par-3 second, Woods found a bunker but blasted out inches from the cup to save par but stumbled on his next swing, sending his tee shot way right and out of bounds by inches on the third hole on his way to a double-bogey six. He was six shots behind, along with Ernie Els (70). "I'm going to have to play very, very good golf on the weekend to catch him, but I'm in a better position this Friday than I was last Friday, so I'll take anything I can get". He made an early bogey, then birdied five times in a 10-hole stretch starting at the par-5 fourth.
He had only two birdie chances inside 30 feet on the back nine (he started on No. 10), and those were from 18 feet.
Stenson settled down from there and ran off three straight birdies, two from close range, the last one from 20 feet on the fringe.
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Hawking appeared on several episodes of the animated sitcom, and his wry humor seemed to be a ideal match for the satirical show. I remember the fuss when his book A Brief History of Time came out, and how it became such a big deal in the UK.
Like most of the day, Woods was unable to get a putt to drop on the 18th after a mammoth 3-wood liner off the tee that left him just 150 yards in.
The disappointment was his wedge play.
"Today was just a bad day", he said.
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.
Woods trailed after 36 holes in four of his eight victories at Bay Hill, including a seven-shot deficit in 2008.
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Australia's Nathan Lyon also was punished for an over-zealous celebration of a wicket in that first test. Cricket South Africa executives Clive Eksteen and Altaaf Kazi pose with fans in Williams masks on Friday.