The head coach was still raging after the game about Lucas Joao’s opening goal in the 64th minute, which he claimed was yards offside.
Substitute Joao netted again with a rasping drive two minutes later and, despite a late Matty Taylor penalty, smashed home after a foul by Josh Onomah on Callum O’Dowda, Wednesday held on to leap to sixth place.
Johnson said: “I have watched the first goal again and it was three yards offside. We knew that at the time and it is so frustrating.
“The first goal in Championship games is so important and today we have conceded it through sub-standard officiating.
“Every day we work with our defenders on squeezing up to get our high press going. An individual has let us down today and it wasn’t one of my players.
“I don’t normally talk about officials, but today it was such a key moment. I think they knew immediately that an error had been made.
“We also have to look at the fact that we failed to take some great chances. But it was a game that could have gone either way and this sort of thing has to stop at our level.
“I thought we played some great stuff at times. We dominated the first half and, while the second was closer, we might easily have won.”
Wednesday boss Jos Luhukay admitted his side had been second best for much of the opening 45 minutes.
He said: “I was not happy with what I saw in the first half. Bristol created several chances, while we had only one or two good moments.
“The second half was much better. We caused them problems, scored two good goals, and after giving away a stupid penalty, defended really well at the end.
“Lucas Joao was having a difficult game after going on as substitute, not always keeping the ball. But he was different after his first goal and in the end was very important to us.
“If he wants to be in the starting line-up, he has to work very hard in training day after day. It’s as simple as that.
“Josh Onomah was too aggressive in giving away the penalty, but I was pleased with the way we defended our lead at the end.
“Steven Fletcher hurt his back and Barry Bannan also picked up an injury. We have had an intensive period and I am happy about the international break.”