The 68,000-seat state-funded stadium hosted its first match in front of a packed-out crowd in Budapest.
Those in attendance were treated to a barnstorming opening to the friendly as all three goals arrived inside 25 minutes, with Cavani and Rodriguez putting the visitors in command.
Adam Szalai scored the first Hungary goal at the ground to halve the deficit but it was a lead they were unable to turn around.
Uruguay were ahead after 15 minutes when Matias Vina's excellent left-wing centre was turned in at the far post by Cavani.
Luis Suarez then teed up Rodriguez, who jinked inside on the left of the box and bent a sublime effort into the far corner.
Hungary reduced the arrears when Szalai coolly slotted home Roland Varga's cut-back with his left foot.
Substitute Maxi Gomez had the opportunity to restore the two-goal cushion when slipped in by Cavani in the 64th minute, only for Botond Barath to make a superb recovery block.
Cristhian Stuani almost bundled his way through in the closing stages, but Denes Dibusz stood up well to his close-range effort.