Robert Snodgrass netted the only goal after 26 minutes to earn the Scots their first win in Group A and dent previously undefeated Croatia's chances of reaching next year's finals in Brazil.
Speaking to Sky Sports News, he said: "It is one of the goodies, don't worry about that - I will not forget this one."
It was Strachan's first win in charge since taking over as Scotland boss, and he put the success down to their build-up.
He said: "The preparation was good, from the players' point of view, I thought that you could not ask more from anyone - be it on the training field or watching videos, what we lacked is fitness, no doubt about it.
"If we had more fitness we could have got up the pitch more - the fact is some players don't play for their club's regularly, some in the Championship - some haven't played for a month, so that is a problem and if we can crack that I know we have got players who can keep the ball longer, and then if we keep the ball longer they have it less."
Strachan insisted that tactics did not play a huge part in their win.
"People talk too much about tactics, the game is a lot to do with players - tactics are nothing without players and being able to close down and pass the ball," he said.
"It makes them feel better about themselves, the fact is that if you play for Scotland and you win, you make a nation happy.
"It's a great result and we must enjoy the result. But when I look at it from a coach's point of view, there are one or two things you would like to brush up on, that's for sure.
"This is something that makes us happy for a while and makes us feel good about ourselves.
"I think sometimes we do take it badly. I remember James McArthur and Robert Snodgrass taking it badly after the Wales game.
"Now they can feel good about themselves and hopefully in the next game there is a spring in their step."
Strachan felt the victory will give Scotland fans some pride back.
He said: "I would think so, I would hope so. You play for your country for pride. It's not for money, it's not for anything else. And you give people pride.
"It's satisfying. The lads played from their hearts. They played as a group, 24 of them trained as a group, which made this possible. So we thank them for that.
"I really feel pleased for everybody who gets on the plane, who travelled, who were watching. When you're with the national side, the group makes the whole country happy."
Strachan picked a number of performances as "heroic".
He said: "There is heroic in terms of defending. There is also heroic in holding the ball up. Shaun Maloney was doing that, and James McArthur.
"Craig Conway and Steven Naismith went on and did well. I think that's as good as I've seen Alan Hutton play. Steven Whittaker as well. As a partnership at centre-half it wasn't bad too.
"Other ones have to get up to a level of fitness to play at this level and that's hard work."