The Watford player tormented Macedonia right-back Daniel Georgievski, who plays for Steaua Bucharest, all night, getting round the outside and putting in a number of dangerous crosses throughout the game.
Anya, who set up a chance for Shaun Maloney when he came off the bench during Friday's 2-0 defeat by Belgium, was even more effective in his second international.
The 25-year-old gave Scotland a well-deserved lead on the hour mark when he collected a pass from Barry Bannan and fired just inside the far post.
Anya told the Scottish Football Association's YouTube channel: "It was amazing. I can't really explain how it felt.
"It was perfect. I was lucky to start and luckily I had a good game.
"When you go on the pitch, you just focus on giving it your all. Sometimes you play good and sometimes you play bad. Luckily for me, things went well.
"But it's not just about me, it was an amazing team performance."
Steven Naismith and Scott Brown could not convert good first-half chances as Scotland dominated and Gordon Strachan's men looked comfortable for 10 minutes after Anya's goal.
But substitute goalkeeper Matt Gilks then pulled off a string of fine saves before Ivan Kostovski levelled when he stabbed home in the 84th minute.
But Maloney conjured up three points that took Scotland off the bottom of World Cup qualifying Group A and into fourth place when he curled a glorious free-kick into the top corner five minutes later.
Anya said: "We were really disappointed to go in at half-time 0-0. I thought we could have had a couple of goals.
"Luckily Shaun came up and did what he did and got us the three points."