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.
Uruguay coach Oscar Tabarez was missing from the sidelines after undergoing back surgery last month, but there was no let up from the South Americans, who moved ahead 11 minutes before the break when Cavani was the beneficiary of a defensive gaffe.
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.
Neither side was able to find a breakthrough in a tight opening and it took a defensive lapse from Hajime Hosogai for the deadlock to be broken.
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.
Despite the setback, Japan enjoyed a spell of pressure towards the end of the half and Yusuke Minagawa saw a goalbound header well blocked.
Junya Tanaka's shot stung the palms of Uruguay 'keeper Fernando Muslera in the first chance of note after the break.
Cristian Rodriguez then tried his luck from distance but his fizzing strike went wide of the left-hand post.
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.
