Jamie Roberts, Scot Williams, George North and skipper Warburton touched down before the break, while Lloyd Burns, Leigh Halpenny, Lloyd Williams and Jonathan Davies all scored tries in the second period to complete a comfortable victory.
"It's something we've worked very hard for over the last four years and it's a great feeling that we've got to the quarters," Warburton told ITV.
"We said we wanted to be ruthless from one to 80 and put in a good performance throughout.
"Credit to Fiji, they kept pushing and had a lot of territory, but our defence held strong and that was a target for us."
Wales qualify after finishing second in Group D, with their only defeat coming against South Africa.