The hosts appeared to be heading for a goalless draw as a combination of the woodwork and goalkeeper Martin Dubravka saved the visitors - who had Robert Mak sent off in the 23rd minute - time and time again.
"I genuinely thought we were going to score a goal. It would have been a huge disappointment if we didn't. That was some performance," Strachan told Sky Sports News.
"Slovakia going down to 10 men shook us for a bit because we had done our work against 11 - and then it changes. But we managed to adapt and we created plenty of chances. The man of the match for them was the keeper - he was magnificent. And the crossbar was good as well!
"We had all these great performances and then the footballing gods decided to give us victory. I'm going to enjoy this for about an hour and a half. We're going to forget about Slovenia until this is over."
Scotland now sit second in the Group F table and will guarantee a spot in November's playoffs with victory in Slovenia on Sunday night.