The former Mexico boss was chosen to replace Alberto Zaccheroni in July, the Italian paying the price for a World Cup campaign that yielded just one point.
However, Aguirre's reign began in defeat in Sapporo on Friday as Japan's opponents clinically claimed victory.
Hernandez - who joined Hull City from Palermo for a club-record fee last week - was one of several substitutes introduced in the second half and made his mark by making sure of the win with just under 20 minutes remaining.
The midfielder's poor touch deep in his own area was picked up by debutant Diego Rolan and his throughball was collected by Cavani, whose shot deflected over Japan goalkeeper Eiji Kawashima before nestling in the bottom left-hand corner.
Junya Tanaka's shot stung the palms of Uruguay 'keeper Fernando Muslera in the first chance of note after the break.
It mattered little, however, as Hernandez made it 2-0 in the 71st minute by sweeping under the legs of Kawashima following a spell of pinball in the Japan box, although the goalkeeper may feel he could have done more to stop the effort.
