Rose, who has won on the PGA Tour at least once in each of the last three years, reached the par-5 16th in two shots and then ran in a 10-foot putt for his eagle. That was part of a stretch of five holes which he played in 4-under and he also had four birdies in a row on the front nine of the Bay Hill course.
His only bogey came at the par-4 11th and he eventually turned in his lowest score in four starts on the tour this season.
John Huh was alone in second after firing a 67 while both John Rollins and rookie Brad Fritsch were at 68.
A win for Woods this week would be his eighth at this tournament and would also boost him past Rory McIlroy into the top spot in the world rankings. Woods fell as low as No. 50 following the personal problems in 2009 that led to a prolonged absence from the tour.
Woods ended a PGA Tour victory drought of more than two years when he won this event in 2012. He is the only multiple winner on the circuit this season.
Mickelson made seven bogeys in a round of 73, leaving him in a tie for 57th place. Brandt Snedeker, who suffered a rib injury after winning at Pebble Beach last month and has been out of action ever since, returned to competition and struggled to a 4-over 76 in the opening round.