A brace from Paddy Madden sent the Dons - who beat QPR in the FA Cup on Saturday - to their first defeat in 2013, with goalscorer Antony Kay dismissed in the second half for an off-the-ball clash with Madden.
The League One game was thrown into doubt but was eventually given the green light when referee Mick Russell passed a late pitch inspection as swirling rain swept across Huish Park.
Johnson said: "We knew that if we could get the game on then we would be up for it.
"It was a big game and we knew that whichever side lost would be very disappointed. We knew that if it could go ahead and we got a positive result, we would find ourselves like a cat amongst the pigeons in the top seven or eight.
"We're getting closer to where we want to be. Of course I wanted the game on as we're the home team, but to be fair, Karl Robinson also wanted it on.
"The referee quite rightly waited a little while to check it was safe, but we all decided to start it and credit to both sets of players. If you walk out there now the pitch looks unbelievable.
"It's absolutely great to have Paddy Madden around - he is an infectious runner, he has a bit of Irish blood in him, and is a great finisher. He shows that week in, week out for us."