Two set-piece goals from Roman Bezjak overturned a first-half strike from Leigh Griffiths, and Robert Snodgrass failed to convert a half-chance in the dying seconds after his late leveller.
Bezjak equalised with a free-header from a Josip Ilicic free-kick and had yards of space to slot home the second after a corner from the same player.
Strachan refused to discuss his own future after Scotland just missed out on second place in Group F to Slovakia – who beat Malta 3-0 – and he declared himself honoured to work with a group of players battling against the odds in tournaments.
The likes of Ikechi Anya, Griffiths, Andy Robertson, Kieran Tierney and Barry Bannan were up against bigger opponents and Strachan claimed that had taken its toll, at the Stozice Stadium in Ljubljana.
“Genetically we are behind,” said Strachan. “In the last campaign we were the second smallest, apart from Spain.
“We had to pick a team to combat the height and strength of the opposition at set plays. Genetically we have to work at things, maybe we get big women and men together and see what we can do.
“But it is a problem for us because we have to fight harder for every ball and jump higher than anyone else.
“Nobody can tell me their technique, apart from one player, is better than any of ours. But physically we have a problem, we are able to battle and get through just with sheer determination and skill and work-rate. And that takes a lot out of you.”
Scotland were unbeaten in their final six qualifiers, but ultimately failed to make up for an indifferent start, with a 1-1 home draw against Lithuania arguably the most damaging result.
“Over the 90 minutes and the last year, it’s been an honour to work with these guys,” said Strachan.
“They gave it a right good shot and left nothing in the dressing room. We’ve been beaten on goal difference by Slovakia, who are excellent.”
Slovenia boss Srecko Katanec said: “I’d like to congratulate my boys. We were clearly the better team the whole game.”