McDowell was one under par through six holes, and 14 under for the tournament, at a soaked Sherwood Country Club in Thousand Oaks, where the golfers played in a steady drizzle.
Woods, who began the day five shots behind McDowell, fell seven strokes behind after the five-time World Challenge winner had yet to record a birdie through eight holes.
Woods did have one bogey, on the par-four seventh hole. His drive landed right of the fairway, his approach fell well short of the green and then, standing in deep, wet rough, Woods left his chip short of the putting surface.
Fowler holed out his 120-foot approach shot at the par-four first hole for an eagle. The flamboyant Fowler, wearing an orange ensemble, hit the shot just past the pin and watched it spin back into the cup.
The World Challenge, which pays $1 million to the winner, is an 18-player invitational event that benefits Woods' charitable foundation.