The visitors started well with Murray testing Wednesday's goal early on, as Glenn Loovens did well to deflect the shot wide.
Reading should have taken the lead on the half-hour mark after Murray was brought down by Giles Coke in the box, only for the Crystal Palace loanee to see his spot-kick smartly saved by Keiren Westwood.
Jacques Maghoma was energetic throughout the first half and was causing the Reading defence problems before the half-time whistle.
It was all Wednesday as the second half went on, with May causing problems down the flanks, but his cross could not be converted by Atdhe Nuhiu, whose first-time strike trickled wide.
It seemed like it was going to be one of those days for the hosts until May's free-kick straight at the wall on the 83rd minute was deflected into the left-hand corner of Mikkel Andersen's goal.
Reading did not look close to finding an equaliser as the final whistle drew nearer and were lucky not to be further behind after a hectic goalmouth scramble was eventually cleared to the relief of the Royals.