Robert Snodgrass, who had scored in the Scots' 1-0 win in Zagreb in June, opened the scoring in the first-half with a header before Steven Naismith added a second after the break when he followed up on a Barry Bannan penalty which had been saved by Stipe Pletikosa.
It was Scotland's first home win of the 2014 Brazil campaign and left them fourth in Group A behind Belgium, Croatia and Serbia with the result enough for Igor Stimac to offer his resignation to the Croatian Football Federation, even though his side are in the play-offs next month.
The Tartan Army was in full voice as the Scots saw out the final minutes which prompted Strachan to say: "That was good.
"I think when you are the Scottish manager you want to make the nation happy and hopefully we have made the nation happy.
"It is a confidence thing.
"You have to dig things out, results, performances, take a few knocks physically and mentally and eventually it gets better and it grows within the crowd and the players feel that as well.
"So the whole stadium becomes more confident about the way we go about our job and it makes us all feel better."
Strachan appeared surprised to hear Stimac had offered to resign, saying: "I will speak to him.
"He is a lovely man, a terrific fellow"
Strachan played down the significance of the final fixture victory, but hailed the efforts of his improving players.
He said: "It might be of no importance whatsoever but it makes you happy for a month.
"It was a nice way to finish the week. It has been hopefully enjoyable but hard work and when you put that kind of hard work in it is always nice to get a win.
"It was a big difference from the first game against Wales.
"But the players have gradually turned it round and they can be really proud of themselves."
Stimac admits he has become fed-up with the criticism that has followed him during the whole campaign.
He said: "It is a bad defeat. I offered my mandate (resignation) to the president of the Croatian Football Federation because of the latest results, because everything that surrounds us.
"Never mind that we qualified for the play-offs, if there is anyone in the federation that does not believe in my work I want to give them the chance to change.
"They will have a meeting to discuss it."