Niki Goneva scored after the break to make things interesting but Dan Biggar's boot saw Wales through, moving them top of Pool A - seven points better off than England, who play second-placed Australia at Twickenham on Saturday.
Full-back Morgan passed his first test, collecting an early clearance and feeding British & irish Lions winger George North, who somehow found his way from the halfway line to the Fijian five-metre.
A period of Welsh pressure followed and the first try arrived on eight minutes, scrum-half Davies sniping around the side of the ruck and Biggar converting.
Fiji at last began to enjoy some possession outside their own 22 and when they were awarded a scrum penalty, Ben Volavola floated his 45-metre effort from near the right touchline between the posts.
A second penalty soon followed for a Welsh infringement in the lineout but, from much closer this time, Volavola sent it wide - Biggar soon showed him how it's done to restore the seven-point lead.
Morgan then made a clean break from inside his own half but when the ball eventually found Taulupe Faletau - winning his 50th Wales cap - he was hauled down just short of the line and pinged for not releasing.
Shortly after North thought he had scored that second try with a pick-and-go from close range but TMO George Ayoub spotted a Bradley Davies infringement in the build-up.
It arrived on the half-hour, though. Quick hands from Biggar sent Tyler Morgan towards the line and when he was halted, hooker Baldwin squirmed over by the left touchline for the fly-half to convert.
Volavola then slotted a straightforward scrum penalty and, after resisting a powerful Welsh rolling maul, Fiji went into the break 17-6 behind.
Volavola fluffed his lines with another simple penalty early in the second half but Fiji soon made amends with a superb team try.
Incisive running had the Welsh defence all at sea and the final offload from Timoci Nagusa fed Goneva to slide over, Volavola converting.
Biggar sent a long-range penalty creeping over the crossbar but Fiji fought back once more through Leone Nakarawa, a customary carry and offload from the giant second row causing havoc.
They duly escaped and Biggar eased the tension with a penalty but Rhys Priestland - his replacement - kicked for touch instead of adding another five minutes from time.
The decision looked to have paid off as British & Irish Lions centre Roberts piled over in the corner, only for the TMO to rule it had been held up, but Wales had done enough.