The Robins boss had come in for some criticism of late as the side had slipped into the Championship relegation places, but they produced a fine performance as goals from Lee Tomlin, Tammy Abraham, Aden Flint and a David Cotterill penalty sealed an emphatic win against their promotion-chasing opponents.
Afterwards, Johnson said: "As far as my players are concerned, I was delighted with every one of them. They carried out our game plan perfectly and it was great to produce such a terrific performance in front of our own fans.
"I hope we silenced a few doubters tonight. I saw this display coming and was confident before the game. Now we go into the international break with real momentum."
The win also came at the end of a difficult couple of days as Johnson's father Gary, manager of Cheltenham and one of his predecessors in the Ashton Gate hot-seat, is recovering in a Bristol hospital from a heart operation, which was carried out by the surgeon father of City defender Joe Bryan.
Johnson said: "This result will be a massive tonic for dad and I want to thank the football world for all the goodwill expressed towards him and our family.
"The heart surgery was performed by Joe Bryan's dad, which we didn't want to talk about before the game.
"I am grateful to him and all the hospital staff. Dad is doing okay and the test results are encouraging."