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Oliver's time was the fastest in the world in this World Championships season, improving on the 13.07 previously posted by both Liu and Cuban Dayron Robles.
Liu, trying to regain his dominance in the event after an Achilles injury battle that scuppered his 2008 Beijing Olympics bid, led Oliver to the first hurdle by the barest of margins.
But the big American had powered past by the third hurdle en route to a blistering early-season time in a race run with a wind of 1.8m/sec.
Liu, who had ended Oliver's 20-race winning streak in the event at the Shanghai Diamond League meeting on May 15, was second 13.00. American Aries Merritt was third in 13.18.
"I'm just happy for a good performance," said Oliver, adding that he knew from the opening strides that he was in a good rhythm.
"I took care of business at the start -- like I didn't do in Shanghai."
Liu wasn't happy with technical flaws at the end of his race, but pronounced himself "very happy" with the time.
Oliver's time was one of a half-dozen world-leading performances produced in the early events of the day.
Sudan's Abubaker Kaki Khamis notched the fastest 800m in the world this year with a victory in 1:43.68. Australian Ryan Gregson posted the fastest mile in the world this year as he won the International Mile in 3:53.86. The meeting was to close with another men's mile, the Bowerman Mile that traditionally caps the action in Eugene, which boasts a wealth of enthusiastic fans of distance and middle-distance running.
Other world leaders in the early events included Ukrainian Olha Saladukha with a winning triple jump of 14.98m on her birthday and American Lashinda Demus -- who won the 400m hurdles in 53.31sec.
"I had a clean race for the first time this year, but it wasn't efficient," Demus said. "I still have work to do."
Nadezhda Ostapchuk of Belarus won the women's shot put with a world-leading 20.59m.
