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."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The Barbarians are one of the top invitational rugby sides with a long history and classic rugby jersey.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Lions get ready for their matches with this green training jersey. It's what the players wear. Get in the Gear!
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
$29.99
The Lions are ready to get their Australia tour underway. They arrive in Perth on Monday.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.