Watson, a two-time winner on the PGA Tour last year, posted a 5-under 67 on another day of low scoring at the TPC-Doral course for a 54-hole total of 199.
"I never thought I would say I'm disappointed with 7-under, but today is definitely one of those days," McIlroy said.
Watson opened his round with an eagle and a birdie and soon found himself four shots in front. He then missed short par putts at both the fourth and ninth holes and fell back into a tie with Rose, who birdied four straight holes around the turn on his way to a 3-under 69.
Watson, however, birdied the 12th and 14th while Rose stumbled late in the round.
Matt Kuchar (66) and Zach Johnson (67) were tied for fifth at 205 and Johnson Wagner (67) was alone in seventh at 206.
Webb Simpson (66) had a chance to be close to the lead, but he double bogeyed the par-4 18th and wound up tied with McIlroy and Woods.