A ruthless first-half display meant they clinched the required bonus point before the break through tries from centres Jamie Roberts and Scott Williams, wing George North and Warburton.
Roberts, substitute hooker Lloyd Burns, wing Leigh Halfpenny, scrum-half Lloyd Williams and replacement centre Jonathan Davies added touchdowns after the break, while fly-half Rhys Priestland booted 13 points and his replacement Stephen Jones kicked the extras for four tries.
"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 said on ITV1.
"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.
Everybody put in a good shift. We played in the right areas. There are a couple of mistakes we can look back on but all in all we can be pretty happy with that."
Asked if there were things Wales could improve on, Warburton added: "A couple of silly penalties and handling errors, maybe trying a little bit too much and getting ahead of ourselves, but we won by 66 points so we can't complain really."
Coach Warren Gatland was pleased to get out of a pool which also included world champions South Africa and a dangerous Samoa outfit.
"That was the target coming here," he said. "We knew we were in a really, really tough group and the first goal was to try and win a few games in this group and make the quarter-finals.
"We've achieved that so it's one game at a time now.
"We worked really hard in the off season, and we said we're too big, too fit, too powerful and hopefully too smart [not to progress], so we wanted to be ruthless.
"I'm really proud; tough conditions, but I thought the boys went out and gave a really balanced performance."
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