With Fiji leading 13-7, wing Eroni Sau was sent off for a reckless swinging arm on Johnny Williams in the 25th minute.
The visitors also picked up two yellow cards.
Hooker Ryan Elias scored twice with Kieran Hardy, Alex Cuthbert, Louis Rees-Zammit and Liam Williams also scoring as Wales finished strongly.
Wales host Australia next Saturday to end their autumn series, having lost to New Zealand and South Africa before overcoming Fiji.
The visitors underlined their attacking threat with two tries from Waisea Nayacalevu and 13 points from Ben Volavola, but will be left to rue their indiscipline with playing periods of the game with 13 men despite leading 10-0, 13-7 and 23-14.
Wales were sloppy and their attacking play lacked cohesion for long periods and were indebted to a sensational late score from wing Rees-Zammit.
Full-back Liam Williams impressed in his first start of the season by running in the final score to complete a late four-try and 24-point spree.
Wayne Pivac's side will be grateful for the victory after two defeats but will require a marked improvement in the final autumn international against Australia next weekend.