Despite suffering mass changes to their squad in the summer, the Whites sit sixth in the Vanarama National League on 14 points.
Ryan Bird's 80th minute penalty was enough for Dover, who have now lost just once in their last six.
Kinnear was rightly thrilled with his young side, and he said: "It's a massive three points. The lads had to dig in, we haven't got the star players like they have and we're not as big a club as Tranmere. I think there were some nerves. A lot of the players hadn't played at this level or in a game like that.
"We always plan to get forward and score goals but they had the ball a lot so when you've got to defend when you've got to defend.
"We played with a lot of forwards out there but that doesn't mean they can't track players. We had some chances to be fair. The two games we've lost are probably the two games we had the most chances.
"The lads stuck in there when we were going through bad patches, I think only one player was with us last season so what can you say? We were really pleased with it."