James Hayter crashed the ball home from close range after 81 minutes to confirm all three points for the Glovers, who had taken the lead through Sam Foley eight minutes after half-time.
The visitors lost Dan Burn to a second yellow card in stoppage time but by then the game was all over.
Johnson's side are fourth in League One, five points off an automatic promotion spot but having played a game less than second-placed Bournemouth, and the Yeovil boss insists anything is possible.
"It's a magnificent achievement for our club, our supporters and our players to get into a play-off position," said Johnson.
"We've still got to keep going though and try and keep touch the team's above us. I wanted us to consolidate and we have. It's nice to do that. It's been a fantastic end to the season now whatever happens.
"In the first half we were absolutely hopeless. I felt most of them didn't turn up. It was 0-0 so that was the only positive.
"All of a sudden I must have touched a few buttons because when we came out after the break we were totally different.
"We looked a Championship team against a team that's hanging on in there. All of a sudden we played our football, we took our chances and scored two goals.
"Sam scored a lovely goal, we know he's got that in him though. James has 15 goals and that's fantastic. He scored a great goal."
- Sports & Recreation
- Dan Burn