Despite a crippling casualty list and being drawn in the toughest pool in the history of the tournament, the Welsh will head to their final Pool A match against Australia next weekend with three wins out of three.
Progress to the knockout stage is not assured for the 2011 semi-finalists but victory for Australia over England at Twickenham on Saturday would secure it.
After first-half tries from scrumhalf Gareth Davies and hooker Scott Baldwin, it looked like Wales would earn the four-try bonus that would have made their position even stronger.
A combination of Welsh fatigue and Fijian fitness, however, meant the home side failed to cross the line in the second half and were reliant on the faultless place-kicking of flyhalf Dan Biggar to usher them to victory.
“Got to give great credit to Fiji, it was very tough. I was knackered in the first half,” Wales captain Sam Warburton reporters at the Millennium Stadium.
“But if someone had said before the start of the tournament we would have three wins from three you would have bitten their hand off for that.
“After the work we did last weekend we couldn’t throw it away.”
Fiji, who had impressed in their defeats by England and Australia, scored the best try of the game through centre Vereniki Goneva and dominated the scrum but were left ruing flyhalf Ben Volavola’s two misses from the kicking tee.
Wales, playing in black and charged with belief after that dramatic win over England at Twickenham, set off at a canter with winger George North storming up the centre of the park in the opening minute.
It took them six minutes camped on the Fijian line before Davies darted through the defensive line to score his fourth try of the tournament under the posts.
The tone had been set for a match both defensive coaches will want to forget but it was not until the 32nd minute that Baldwin forced the ball over the line for the second try from close range.
That gave the Welsh a 17-6 halftime lead but the Fijians stuck to their task and a searing break on the counter-attack by Asaeli Tikoirotuma set up Goneva for a brilliant converted try which cut the deficit to four points after 48 minutes.
Biggar gave the home side a bit of breathing space with his second penalty but the Fijians came back at them and the Welsh were clearly starting to feel the effects of a five-day turnaround from the Twickenham match.
Again, it was Biggar who soothed Welsh nerves with a penalty to give his side a 10-point lead going into the final 10 minutes and it proved enough.
“I’m very proud of our team from one to 15,” Fiji coach John McKee said.
“The effort and hard, hard work that went into the game and to go so close is frustrating but I can’t fault our team for their work and effort.”
(Editing by Ed Osmond)