Johnson was on hand to grab his first goal of the season minutes after Youga had seen red for a second bookable offence.
Bradford boss Colin Todd chose to keep faith with his new-look front pairing of Bruce Dyer and Billy Paynter, who both scored on their debuts against Nottingham Forest last weekend, while Gillingham welcomed back Mark Bentley from a ban and switched Matt Jarvis to a striker's role.
After a tentative opening, the game finally sprung to life in the 22nd minute when Bradford's recent signing Omar Daley surged forward from his own penalty area, exchanged passes with Dyer, before crossing to Marc Bridge-Wilkinson who forced a point blank save out of Gills keeper Romain Larrieu.
Gillingham responded with Gary Mulligan forcing Donovan Ricketts to tip over, but their task was made easier when referee Andy Hall showed Youga his second yellow card after carelessly tripping Bentley.
And deep into first-half injury-time, Gillingham made their advantage tell when Johnson rose to head home Clint Easton's corner from close range for his first goal of the season.
The Gills should have doubled their advantage with an almost identical play shortly after the break when Bentley was given the freedom of Gillingham to meet Easton's corner but headed tamely at Ricketts.
Andrew Crofts twice blazed over from decent positions after being played in by Nicky Southall and substitute Guylain Ndumbu-Nsungu respectively.
And in the 78th minute Jarvis burst clear only to see his shot from the edge of the area strike Ricketts' far post and hacked clear to safety.
Danny Jackman fired a long-range effort narrowly over late on as Gillingham managed to secure all three points without any late scares.