Barkley finished 19-for-20 for 298 yards and Woods hauled down eight passes overall for 132 yards as the Trojans (6-1, 4-1 Pac-12) won their fourth straight.
D.J. Morgan, Marqise Lee and Xavier Grimble each made a scoring grab in the victory.
The Buffaloes (1-6, 1-3) were set back by three interceptions surrendered by Jordan Webb, who completed just 19-for-36 passes for 214 yards.
Celebrity Families of 2014 [PHOTOS]