Wales went over again in the 15th minute but Cameron Winnett’s effort was chalked off after a dangerous clear-out from Tommy Reffell in the build-up that earned the loose forward a yellow card.
Despite the disadvantage, Wales were awarded a penalty try four minutes later when Elia Canakaivata tried to pull down their driving maul, earning himself a 10-minute stay in the sin bin.
Fiji’s woes were extended in the 21st minute when key winger Semi Radradra was cautioned, upgraded soon after to red, for a high hit on Winnett.
Radradra was sent off as World Rugby trialled a new red-card protocol, enabling Fiji to bring on a player after 20 minutes had passed.
Despite being down to 13, Fiji scored with a break from Muntz, who cut through the middle and accelerated away from the defenders for a brilliant 24th-minute individual effort that reduced the home lead to 14-10.
Two minutes into the second half, Muntz reduced Wales’ lead to a single point when putting over a penalty, and then gave Fiji a 16-14 lead with another successful kick in the 49th minute.
Tuisova powered over for Fiji’s second try on the hour mark, as a box-kick set up a powerful attack at the Welsh line and silenced the home crowd.
Substitute Jac Morgan, making a first appearance since last year’s World Cup, helped to set up Wales’ third try five minutes later with a carry that stretched the visiting defence and once the ball went down the line allowed Bevan to go over and reduce Fiji’s lead to 21-19.
But it was the Pacific islanders who finished stronger, botching a chance for a third try with an errant pass six minutes from the end.
They had, however, a penalty advantage in the build-up and Muntz put over the close-in kick to leave Wales scrambling unsuccessfully in the closing stages to try to snatch the win, with Ryan Elias spilling the ball in their last attack.
Wales’ last victory was a 43-19 triumph over Georgia at last year’s World Cup, the game after they had also thrashed Australia.
Since then, they’ve been eliminated from the tournament by Argentina, suffered a Six Nations whitewash and mid-year losses to South Africa and twice to Australia.