Former world junior champion Baghdatis, a hardcourt specialist, had lost his previous three matches on grass and was bidding for his first win since the second round of the French Open in May.
"It's been two months and that's a long time (without winning)," the 25-year-old told reporters after reaching his fifth ATP World Tour quarter-final of the year.
"I came to LA knowing I wasn't going to be playing my best tennis.
"I came with this mentality of fighting for every match. I felt really bad on court and it's good to win a match when you feel bad. The positive attitude is there, which is very important, looking forward to the next one," he added.
Baghdatis, who clinched his fourth ATP title in Sydney in January, fired down 11 aces on the stadium court at the LA Tennis Center before sealing victory in one hour 50 minutes.
He will next meet sixth-seeded Serb Janko Tipsarevic who scraped through a first-set tiebreak 11-9 before beating Indian qualifier Somdev Devvarman 7-6, 6-2.
In the first match of the day, German veteran Rainer Schuettler became the first player to reach the last eight by battling past American Robby Ginepri 6-3, 3-6, 6-4.
At 34 the oldest player in the draw, Schuettler converted three of his seven breakpoint opportunities to set up a possible encounter with second seed Sam Querrey in the quarter-finals.
Big-serving American Querrey, who beat Australian Carsten Ball in last year's final, was scheduled to play South African qualifier Kevin Anderson in Wednesday's late match.Murray top seed in LA after Djokovic withdraws
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