Warriors edged out their old rivals for a 20-16 win and will be looking for a repeat showing on Saturday.
Speaking in The Scotsman, Johnson said: "The win in the first game obviously puts us in a good spot. We have reviewed certain things that we still need to improve on. Last weekend was one of those games where whoever won the physical battle was going to come out on top and I think we did that.
"Both teams weren't holding back. But, in terms of the brutality of it all, I think we won the battle."
Edinburgh have enjoyed a good run of form and went into last weekend's match as favourites but Johnson feels a point was proved.
"The media have obviously been talking Edinburgh up this season, but it's all good... I didn't think we had been playing all that poorly the few weeks prior to facing Edinburgh, even though everyone was saying otherwise," he added.
"At crucial times, our decision-making maybe wasn't what it should have been. But it's not like we were getting completely turned over.
"When the game was on the line against Edinburgh, we really stuck to our task and it went our way in the end.
"Our game is based on winning races into contact. We're known as a flair team but, when we need to, we can also grind it out. The fundamentals are here. Our forwards especially have been outstanding over the past few weeks.
"That's probably put more onus on us backs to be honest and we've just tried to make smarter decisions at crucial times in the game."
Looking ahead to the return fixture at Murrayfield. Johnson said: "It'll be a great contest come what may in front of 20-odd thousand people. It'll be a step up again.
"Edinburgh will obviously be hurting and won't want that to happen again. But the same applies to us in terms of our focus. We're not just going to go out there and just assume that we've won the 1872 Cup. We'll look for areas to improve and go again."
With four defeats in the PRO14 and two in Europe, it's been a below par start to the new season for Warriors. The World Cup in Japan was always going to present a challenge with so many Warriors involved, but Johnson is now looking to build on the Edinburgh win.
"We're a very close group," said the 26-year-old. "There definitely weren't any sort of crisis meetings or anything like that in the camp.
"That's the challenge for us now - we need to back it up. We need to find that level of consistency in our game.
"A bad team doesn't go over to La Rochelle and win. We knew we just had to improve our key decision-making if we wanted to start getting results.
"We know Edinburgh are going to come at us really hard. It'll be up to us to back it up from last week and win that physicality battle again."