Kevin Naiqama went over for a try of his own three minutes later after reacting quickest to a deflected kick to go past a defender and touch down.
Fiji had another try from Petero Civoniceva disallowed before Ireland eventually got themselves on the scoreboard, Tyrone McCarthy making sure that their increasing pressure paid off by going over in the corner.
The Pacific Islanders kicked a penalty just before half time and were back in try-scoring form shortly after the break when they turned the ball over deep inside the Irish half, allowing Tariq Sims to score.
Fiji secured the win with two tries in quick succession just after the hour mark, Uate going over for his second of the match just a minute before Korbin Sims took advantage of a defensive mix-up from a high kick to score their fifth try.
There was still time for Uate to complete his hat-trick with a try in the 71st minute, while Ireland provided a late flurry with two tries, from Damien Blanch and James Hasson, in the final four minutes to make the scoreline more respectable.