The match was close throughout, as Hawkins refused to be shaken off.
Eventually Welshman Stevens crossed the winning line though, sealing a 13-11 success, aided by a break of 57 in a final-frame victory march, and the 34-year-old will face compatriot Ryan Day in the last eight.
"It was very tough," Stevens said. "I stuck in there yesterday [Sunday] after being 3-0 down. Barry played very solid in the opening few frames and I was under the cosh all the way really.
"Today [Monday] was very tense, both of us kept missing the balls. it was tough but it's nice to get back in the quarter-final again."
Stevens, finalist in 2000 and 2005, last reached the quarter-finals in 2007 and failed to qualify for the World Championship in 2009 and 2010.
He was knocked out in the first round last year, but is enjoying the tournament in Sheffield this time.
Day is next, competing at the last-eight stage for the third time, having last gone so far in 2009.
"It's nice to see us back in the quarter-finals again," Stevens said.
"Ryan's had a tough time the last couple of years. It's nice to see him come back, especially with a good first-round win over Ding (Junhui).
"I'm very pleased he's got to the quarters but hopefully he won't get any further."