They are now three points behind fifth-placed Livingston, a situation that had appeared unlikely earlier in the season when they propped up the table.
"People wrote us off far too early," said Wright, who is now preparing for a weekend trip to face sixth-placed Hibernian at Easter Road.
"Whether we get into the top six or not, the players have proved the people who wrote them off wrong.
"And they deserve loads of credit for the way they have pulled the season around.
"We will go to Hibs and our best away performance came against them there earlier in the season."
It might not have been easy on the eye but Saints took the points with a rare moment of quality in testing conditions.
"Keeping a clean sheet against a team who ask you so many questions with long throws and balls into the box was a big test for our centre halves and fair play to them," said Wright.
"And it was a great finish from Callum.
"We are three points behind Livi now so we have given ourselves a chance.
"They have had a lot of plaudits this season and rightly so because they have been very good."
Livingston created the clearer cut chances in the first half, with Steven Lawless testing keeper Zander Clark from 25 yards before sliding another effort wide of the target.
But manager Gary Holt lamented a late lapse in concentration which led to Saints' winner.
Now, with three games left before the split, Livi have a fight on their hands to hold onto their top-six status.
"St Johnstone got too easily down the side of us and ultimately we didn't defend it," said Holt.
"It was a good finish but from our point of view it was poor defending.
"There are three games left and wherever we end up we'll end up.
"Some people think you can't be critical because we have had a good season.
"But if we don't win the next three games we might not be in the top six."