Watson shot a 5-under 66 at the TPC-Scottsdale course for a two-day total of 12-under 130.
That was the same score posted by Matt Jones, an Australian who went to school at nearby Arizona State University and whose best showing on the tour has been a runner-up finish last year at the Wyndham Championship.
Watson bogeyed two of his final three holes to miss out on a chance to lead the tournament by himself.
Patrick Reed, a winner two weeks ago at the Humana Challenge, was among those at 134. Scott Stallings, who won last week at Torrey Pines, missed the cut.
Y.E. Yang, who shared the first-round lead with Watson, shot a 2-over 73 and slipped into a tie for 15th at 137. Phil Mickelson was another shot back after firing a 67. Mickelson was questionable for this tournament after developing lower back soreness at last week's tourney, but he put together a bogey-free round Friday.
Westwood shot a 75 with his round including a sudden loss of concentration at the par-5 13th hole. He had an eight-footer left for birdie on that hole, but it took him four more putts from there. Westwood missed the cut by a single shot.