Westwood, whose only PGA Tour victory came in New Orleans at the 1998 Freeport-McDermott Classic, had been well poised at the halfway mark after a second round of four-under-par 68 at Redstone Golf Club in Humble, Texas.
That had left the 36-year-old European number one just two shots behind 36-hole leader Bryce Molder of the United States.
The world number four's third round was not going according to plan, however, after he finished his front nine with a double bogey, par, bogey and triple bogey to go out in 41.
Westwood improved with birdies at the 10th, 12th and 13th holes but he was still two over for his round and five under for the week as American Joe Ogilvie was making his move at the top of the leaderboard, 11 under overall and four under for his round with three holes to play.
Ireland's Padraig Harrington was the leading European player in the tournament at six under par but level par for the day with one hole to play after a rollercoaster round that featured an eagle, four birdies, four bogeys and a double bogey.
Overnight leader Molder was also level for the day at nine under for the week having played 14 holes, two shots behind Ogilvie, with two more Americans, Vaughn Taylor and Anthony Kim, also giving chase at nine under, both with two to play.
Former Open champion Justin Leonard and South Africa's Charl Schwartzel had the rounds of the day in the clubhouse so far, both carding 67s to move to six under.
England's Justin Rose bounced back from a level-par round on Friday to shoot a 70 and move to four under alongside Sweden's Fredrik Jacobson, who shot a 69.
World number three Phil Mickelson rebounded from a far worse second-round 76 with a 70 to get to one under while Denmark's Soren Kjeldsen's 73 left him at even par for the week, alongside Germany's Alex Cejka (71) and Scotland's Martin Laird, who fell away with a third-round 76.
South Africa's Ernie Els was also going backwards having shot a 70 then a 74 in the wake of back-to-back wins in Florida at Doral and Bay Hill. Els posted a 75 today to slip to three over.
