Byron Webster, Joe Ralls and Kieffer Moore all saw red in a stormy second half at the Madejski Stadium.
"I've no complaints on the day. The sendings off were not for viciousness," Johnson said.
"The first [Webster] was just misjudgement and the second and third were two young lads who were a bit overzealous in the tackle. If you do that, you're going to get a red card.
"It's a lesson learnt for all three of them. They put their team in trouble but they didn't mean to. It was more a question of mis-timing.
"With only eight players, I was very proud the way we defended. There is a passion among all the players and they want to do all they can to keep Yeovil Town in the Championship. We'd like to stay there as long as we can and if we survive, it will be the club's biggest ever achievement."
The first half contained little suggestion of the disciplinary madness to follow after the break.
Yeovil dominated a poor Reading side and deservedly went ahead in the 20th minute. Luke Ayling slung over a free-kick and, with Reading goalkeeper Alex McCarthy racing from his line but not reaching the ball, Shane Duffy had an easy task to nod into the unguarded net.
Reading gradually improved, with Pavel Pogrebnyak having a close-range header disallowed because of offside, and they went on to batter Yeovil in the opening stages of the second half. But leading scorer Adam Le Fondre twice wasted good opportunities - first with a weak header, then with wayward shot high over the crossbar.
Yeovil's red mist descended in the 66th minute, when Webster was sent off for pulling back home substitute Royston Drenthe.
Le Fondre took the resultant spot kick but Yeovil keeper Marek Stech saved well. However, just seconds later, Stech appeared to score an indavertant own goal as Tom Lawrence's attempted clearance bounced off the Czech's back and over the line.
Ralls received Yeovil's second red, for an ugly tackle on Garath McCleary, and Moore the third, in the 88th minute, after he had picked up two yellows in as many minutes - first for kicking the ball away, then for fouling Chris Gunter. But battling Yeovil held on, even after more than seven minutes of stoppage time.
Stech was the hero time and again, making spectacular stops to deny Danny Guthrie and Pogrebnyak as Reading piled on the pressure against their depleted opponents, who moved off the bottom of the Championship with their gallant draw and are now four games unbeaten.
Reading manager Nigel Adkins, whose side dropped out of the play-off places and into seventh, said: "We started the game very well but yet again conceded a poor goal and gave our opponents something to cling on to.
"Then we just started lumping the ball forward.
"When they were down to eight, we got in a lot of crosses and created a lot of chances. We only got one point, when we should have had three, but credit to Yeovil. We've got to pick ourselves up and go again. We're still in the mix."