Saints fell behind after just eight minutes at Pittodrie to a goal from Ryan Jack but had chances to get back on terms.
However, they could not take any of them and to cap a frustrating afternoon they had substitute Steven MacLean sent off late on.
Wright said: "We did deserve a draw at least, I don't know how we didn't get anything from that game.
"We totally dominated the second half and we have got to take the positives out of the game, particularly in the second half.
"It was a soft goal to concede, it goes through a crowd of players then Alan Mannus slips and is a bit unsighted. But the shot should not have come in the first place. We should have dealt with it in the wide areas, but credit to the players who responded well to going behind."
MacLean was dismissed in the 90th minute after picking up a second booking, and Wright admitted that he was baffled by referee Willie Collum's decision to send him off.
He said: "It was a mistimed tackle by Stevie, but if we're going to be giving second bookings for challenges like that then we're going to end up playing seven-a-side.
"I think if he (the referee) took a few seconds to reflect on the challenge, he might not have given it.
"The fourth official said that it was a reckless challenge, but if that was reckless then so was the challenge moments later by an Aberdeen player, all we're looking for is a bit of consistency."
Aberdeen manager Derek McInnes admitted that St Johnstone created the better chances in the game.
He said: "I anticipated it being a hard game. The onus was on ourselves to start well and get ourselves in front. I felt we moved the ball around quickly and with Ryan Jack getting the early goal it certainly helped.
"We've played St Johnstone three times prior to today and they've created very little if at all against us, we've really kept their chances to a minimum.
"But today they created a lot, got themselves into good positions and I'm sure they'll be kicking themselves that they haven't made more of that because they certainly had two or three of the better chances."