Wright was taken to hospital this week with stomach pains and underwent a gallbladder operation on Thursday, leaving the team in the hands of number two Callum Davidson and coach Alex Cleland.
And the players, who were selected by Wright from his sickbed, responded by securing a victory that widened the gap between them and their rivals in the table.
Steven MacLean struck the hosts' opening goal before Ryan McGivern's own goal doubled their advantage in the second half.
Hibernian's hopes of fighting their way back into the game disappeared when Danny Haynes was sent off moments later for an alleged stamp on former Hibs midfielder David Wotherspoon.
St Johnstone took the lead in the 17th minute when Wotherspoon's corner was flapped at by Sean Murdoch, in the team after regular number one Ben Williams' wife went into labour ahead of kick-off, and MacLean got the final touch in a goalmouth scramble.
The striker should have got a second 11 minutes later but incredibly missed as he volleyed over from just three yards out.
Hibs knew they needed a massive turnaround in the second half if they were to maintain their top-six dreams, but things went from bad to worse for the Easter Road outfit after the restart.
Michael O'Halloran did the damage with a surging run to the by-line, but there was a touch of fortune as the striker's cutback was knocked into his own net by the retreating McGivern.
Two minutes later and it was all over for Hibs. Haynes connected with Wotherspoon in a jumble of legs on the touchline and the referee reacted by showing the striker a straight red card.
It was then a matter of how many St Johnstone would win by. MacLean headed over in the 63rd minute and six minutes later Murdoch did well to hold Dave Mackay's shot and a Stevie May effort.
With a ricochet from visiting defender Michael Nelson pinging off the outside of the post, and Murdoch repelling a May shot before Gary MacDonald bundled another effort just wide, it was a surprise that Saints did not triumph by more.