Higgins, who becomes the first player from the top eight seeds to make it to the last eight, raced into a five-frame lead against the number 12 seed, with breaks of 57, 53 and 74 in the first three frames before Hawkins finally began to find his form.
40-year-old Hawkins pulled back a frame with a century break but Higgins wrapped up the win with a 59 break in the final frame to set up a quarter-final meeting with Ali Carter.
Earlier in the day, defending champion Trump crashed out 6-3 against Shaun Murphy after the ninth-seed produced a spirited comeback from 3-2 down.
John Higgins and Jimmy White can't stop laughing - "This has done your swede in!"
Trump made three century breaks to pull into a convincing lead before Murphy hit back with breaks of 77 and 52 to ease through and book his quarter-final slot with Joe Perry.
The departing champion and world number one said: "He [Murphy] just dug in.
"I didn't really do that much wrong. It's frustrating really because I felt like I was in control of the game and he was just hanging on.
"It felt like I didn't play that badly and 6-3 was maybe a little flattering for him."
The first round concludes tomorrow when Kyren Wilson takes on Jack Lisowski and Mark Williams faces Stuart Bingham.