Lee Stevenson scored the only goal of the game on the stroke of half-time, when he cashed in on some charitable Exeter defending to fire a low shot across goal and into the far corner.
But it was Mansfield's defensive display that proved the backbone for the win as they worked stoutly in a second half in which they repelled an Exeter onslaught.
"I can't remember ever having to defend like that," Murray said.
"It was backs to the wall. You have seen some tired legs out there and it was just sheer determination, team spirit and a hell of a lot of desire to see it through.
"The ball just coming back and in the end we just went for bodies behind the ball to try and see it through.
"We still need to be a lot better with the football, we know that, we are working on it, but you can't argue with three points away from home.
"I'm not going to lie and say it was a great performance because it wasn't the best second half.
"They were getting on top of us, but we did what we had to do.
"It is a massive three points, words cannot express how big they are.
"It was a huge game for us and the next building block to move on. It was a long journey down here and we are delighted."
Exeter manager Paul Tisdale felt his young side were unlucky in defeat, but are close to becoming a good team.
"It was a really good game of football and a game that deserved more than the one goal it got, he said.
"As a performance, I think we were on the betters side and I don't think we played poorly and as the players feel, I feel really frustrated that we didn't even get a draw.
"But I suppose that happens when a team hangs on and we have seen it once or twice. It was the same for us at Cheltenham really.
"We are an inch away from being a good team, but it is going to be a case of a step forward, a step back, two forward and one back.
"That is how it is going to go, but I was really pleased with a lot of the players.
"Although I have made it absolutely clear to them that you can't call yourself a good side.
"You are a losing side if you concede a goal and can't score.
"That is the vital part of the game, so how do I feel? I feel frustrated, I feel we were unlucky, but we didn't put the ball in the net or keep it out at our end. There's not much else for me to say really."