Saints could not build on Scott Tanser's eighth-minute penalty and found themselves trailing after a dominant first half thanks to Billy McKay's double, the first a rebound from his own saved penalty.
Saints had a glorious chance to level when Chris Kane was sent clean through but the striker shot straight at Ross Laidlaw, prompting Wright to boot a bag on the touchline in anguish.
The result leaves County top of Betfred Cup Group B on nine points with Saints on zero and with no chance of reaching the second round.
Wright said: "It was as good as we have played at home for a while, on the front foot, everything we did was positive, creating chances. There is no question we should have been 3-0 up. They were causing us no problems.
"The penalty is soft. Jason (Kerr) is probably a bit naive, he has got him going where he wants, but there is slight contact. We paid the penalty but we didn't defend the first ball and the build-up to the second goal is similar with centre-halves going into areas they shouldn't to try and win balls.
"It's unbelievable we came in at 2-1 when they never laid a glove one us, but fair play to them, they weathered the storm, had that little bit of luck that we didn't take our chances, and we think we had a big shout for a penalty on (Danny) Swanson. Certainly, if their one was a penalty, the one on Swanson was a penalty.
"They did better at the start of the second half, then it became a bit scrappy, but we have the best chance of the second half when Kane is through and we don't put that away. And there is numerous good positions we get into.
"We created a load of chances against a side that's going to be in the same league as us, so we have somehow got to get the right personnel in to convert those chances."
When asked about the prospect of re-signing Stevie May, who is free to leave Aberdeen on loan or permanently, Wright said: "We are actively looking to bring at least two players in. I have no update on how things are moving on a striker or a defender."
The win was a confidence boost for County ahead of their return to the Ladbrokes Premiership.
Co-manager Steven Ferguson said: "We are not reading too much into it, it's so early in the season but as far as the steps we want to take, it doesn't do us any harm."
Ferguson offered little clarification following reports the club had accepted a bid from Rotherham for midfielder Jamie Lindsay.
"Over the next 48 hours we'll be able to have something for you," he said.
"At the moment there is nothing regarding Jamie Lindsay. You know as much as I do."