Westwood, in danger of falling to third in the Official World Golf Ranking with Luke Donald resuming in joint second spot, went to the turn in a three under par 33.
It could have been better, but he three-putted the short ninth and with that slipped to 20th place on five under, eight adrift of overnight leader Dustin Johnson.
Among those Westwood had moved ahead of was Tiger Woods, whose winless stretch was certain to reach 19 tournaments going back to November 2009.
Also two under when he resumed, Woods found birdies at the first, fifth and seventh, but driving into sand on the sixth led to a bogey five.
Phil Mickelson, who has failed to win any of his last 22 tournaments, remained even further down the 66-strong field. With five holes to play he was one over and in a tie for 47th.
Johnson led by two from Donald and fellow Americans Matt Kuchar and Nick Watney, while Northern Ireland's Rory McIlroy was only three off the pace in fifth place and Scot Martin Laird, Italy's Francesco Molinari and Dubliner Padraig Harrington just one further back.
Westwood just missed out on an eagle at the tenth, but back at the 529 yard first Harrington made one from 15 feet and leapt into a share of second place.
First and second-round leader Hunter Mahan was on the same 11 under mark after getting up and down from sand there, but Laird three-putted for par and remained nine under.
After the high of the eagle Harrington crashed almost certainly out of the hunt by going in the lake twice on the 438 yard third.
He did hole a 14 footer, but it was for a triple-bogey seven and, with Johnson two-putting the first for birdie, he went from two behind to six adrift.
Donald and McIlroy also made four on the opener, Donald playing yet another brilliant bunker shot to add to his collection, and Laird's five foot birdie at the second kept him in contention.