The Scottish striker saw his free-kick deflected in to earn the Owls all three points.
But the hosts had goalkeeper Keiren Westwood to thank for keeping them in the contest before May’s intervention.
First the former Sunderland goalkeeper saved Glenn Murray’s first-half penalty, then blocked Jake Cooper’s close-range header after the break.
Westwood also parried Nick Blackman’s 30-yard effort as the Royals had the better of the chances.
At the other end, Wednesday were denied a couple of penalty shouts themselves while Atdhe Nuhiu fired wide from May’s cross.
Owls coach Stuart Gray said: “It’s nice to put that one to bed because people keep saying that we haven’t won here since March 26 in the league so they’ll have to think of another question now next week!
“I just felt that the penalty save was probably the defining moment in the game. It gave us a lift and it gave the fans a lift. We seemed to then pick up a bit of momentum.
“Credit to Reading because they closed us down and pressed us. They forced errors, but the players responded.
“We started the second half very well and probably deserved the win in the end.
“We could probably have had two more penalties. The referee perhaps needed a little bit of help from his linesmen.
“Stevie May’s free-kick took a deflection and the ball ended up in the bottom corner. We earned that break today, I thought. We went to Bolton last week, had 25 goal attempts and didn’t get a break.”
Reading boss Nigel Adkins was pleased with his team’s performance in testing circumstances after having to make two last-minute changes to his starting line-up.
He said: “I thought we did well in the first half and kept the crowd quiet.
“We had to make two enforced changes this morning. Jake Cooper came in to make his starting debut at centre-half because Michael Hector picked up a head injury the other day so we had to follow the FA protocol. Jake has come through the academy and I thought he was top drawer.
“Mikkel Andersen made his debut in goal because Feds (Adam Federici) was just feeling his calf. He had one shot on target to deal with and unfortunately it was deflected past him for the winner.
“It was a good Championship fixture. We had some good chances but Westwood made some good saves. Unfortunately, he saved the penalty with his feet.
“Was it a penalty when Alex Pearce knocked the ball back and the ball struck the defender’s arm? It’s fair to say the referee had an interesting afternoon.”
Reading’s takeover by an Asian consortium was confirmed this week after some unrest following the exit of Anton Zingarevich from the board this summer.
And Adkins is delighted the club have now resolved their situation off the field.
“I’m really excited about the future,” he said. “The stability is now here because for a good 18 months I would suggest it hasn’t been there. There are very exciting times ahead.
“Instead of having to sell players we might be able to bring some in.”