In a tight encounter both sides scored a try a piece but it was the boot of Wales fly-half Dan Biggar and a dominant scrum led by the Welsh front-row of Paul James Richard Hibbard and Adam Jones that won the day for the Swansea based side.
"The scrum set the tone," said Johnson, after a hugely physical encounter in Swansea.
"Sometimes when you have such a dominant scrum, the flow of the game seems to cease a little bit. So Daniel did really well to convert.
"Sometimes you can have a dominant scrum and get a lot of penalties but you don't convert your chances. Daniel's done wonderfully well to convert them.
"I'm happy and relieved but not delighted. You get delighted when you know that you've done everything right. We were rusty in some areas of the game. We've had a sensational lineout all year and tonight it wasn't at its best. That didn't give us the front foot but, I'm being greedy, I'll take the win.
"I think the effort was first class. It was brave. Since the autumn internationals, we've really only had one training session together because of the inclement weather so we're a little rusty as a group. But we're happy and I'll sleep a bit better tonight, which will be nice."
Saturday's vital victory for the Ospreys ensures Pool 3 remains wide open with two rounds left to play.
Munster, RC Toulon and the Ospreys all still harbour serious hopes of reaching the knockout stages and Johnson admits it would take a brave punter to pick a winner.
"If you give me a coin, I'll throw it in the air, it's a bit like that," added Johnson when asked how he saw the fate of the group.
"I think we'll still be talking about this pool right up until the very last game. And I think everyone knew that when the draw came out.
"We play Toulon here so that's a good chance for us to control the pool but we've got to go to Irish and take some points too. It's going to go down to the wire.
"We knew when we looked at the draw at the start of the year, there was going to be a few bumpy roads in this pool. While we've had opportunities to take some games that we thought we could have, I think other sides in the competition will be saying the same thing. On balance, we're not too far off where we need to be."