Nemia Kenatale and Leone Nakarawa, plus two penalty tries, helped the Pacific Islanders lead 26-10 at half-time, earning a bonus point before the break.
Uruguay had not scored a World Cup try since 2003, but Carlos Arboleya and Agustin Ormaechea crossed in each half.
Fiji eased away in the final quarter with three more tries, as Uruguay were reduced to 14 men as Ormaechea saw red.
Uruguay now travel to Manchester for their final pool match on Saturday against hosts England, who have also already been eliminated from the tournament.
