He was Johnson's assistant as Yeovil won promotion from the Conference to the Football League, and then again last season when they topped League Two ahead of Scunthorpe.
Thompson has presided over a 1-0 home win against Port Vale, a 4-1 defeat at Southend and last week's 4-3 win at Scunthorpe.
He told BBC Radio Bristol: "It's a difficult task to follow Gary, but that would be the same for anyone coming in. I'm willing to take on that challenge.
"We'll introduce some new ideas and I feel we will make a success of it."
Yeovil chairman John Fry added: "We're very pleased with what he's done, his knowledge of the players, and quite clearly he has the support of the fans.
"Of course he's got things to learn but he will do so with our support behind him."