Ikechi Anya and Shaun Maloney struck for the Dark Blues in Skopje to lift the nation off the bottom of World Cup qualification Group A.
Strachan's men were presented with a heavily rutted surface at the Philip II Arena but impressed with their slick attacking moves.
Yet the the Scots looked like they would have to settle for a draw after Jovan Kostovski had cancelled out Anya's stunning opener.
But Wigan playmaker Maloney came up with an even more audacious winner as he struck home a 20-yard free-kick two minutes from time to secure the victory.
Strachan said: "I'm extremely happy, for the players first of all. To put that work in and play like that and then get hit by a late equaliser was harsh. But we came back again to score and we deserved it.
"I thought they were terrific tonight.
"A draw would have been hard to take. I was coming in on the team bus tonight and I was trying to think how we would manage on a surface like this. How do we win a game of football on this?
"I thought a 1-0 would have been great. But to win it like that was beyond what I would have dreamed."
Strachan handed Castlemilk-born Watford winger Anya his first start for the Dark Blues just four days after he made his debut in the 2-0 defeat to Belgium in Glasgow.
And the 25-year-old gave a glimpse of his talent at Hampden with a couple of pacey runs but his display against the Macedonians was a step up again.
He tormented opposition right-back Daniel Georgievski for much of the first half but proved his quality by slamming under Tome Pacovski on 59 minutes from an almost impossible angle.
But Strachan also hailed the performances of his attacking colleagues Steven Naismith - promoted from an unused sub against Belgium to lead the line as a solo striker - and Maloney.
"Ikechi was good, wasn't he?" said the Scotland boss.
"I thought he was terrific. He brought us that wee extra turn of pace, was a threat and made them worry up front.
"But I thought Naismith up front was terrific. Maloney too. We didn't have a failure."