Needing a victory to guarantee second place in Group F and a spot in the play-offs for the remaining World Cup berths, Scotland took the lead through Leigh Griffiths' first-half strike in Ljubljana.
But Roman Bezjak netted twice for Slovenia in the second half to leave Scotland on the brink.
Robert Snodgrass set up a dramatic finish with his late equaliser, yet despite a red card for Slovenia's Bostjan Cesar in stoppage-time, there was no miracle escape act for Scotland.
Strachan's men finished level on points with second-placed Slovakia, who won 3-0 against Malta to go above the Scots on goal difference.
Scotland are still waiting for a first major finals appearance since the 1998 World Cup in France.
Strachan refused to be drawn on talk that he should resign after Scotland's failure.
"At this moment, I am the last person thinking about that," he said.
"I have got to look after the players. We are hurting, no one is hurting like the players.
"There shouldn't be any talk about what I'm thinking at this moment in time because I am just looking after them.
"I have never been in a dressing room as silent as that."
Slovenia, who were also eliminated by the draw, had not conceded a goal at home in the entire campaign, but Celtic forward Griffiths ended that run in the 32nd minute.
James McArthur headed the ball in the Slovenia penalty area after Darren Fletcher's cross was half-cleared and Griffiths drilled home from a tight angle.
Scotland couldn't capitalise on that crucial breakthrough however and Craig Gordon had to make a good save after half-time to keep out Rajko Rotman's drive.
Slovenia's pressure was rewarded in the 52nd minute when Bezjak headed home after poor defending left him unmarked at a free-kick.
Scotland were losing control and the hosts went ahead in the 72nd minute.
Josip Ilicic's corner was laid back to the unmarked Bezjak, who rolled the ball into the net from 12 yards.
Fletcher had a glorious chance to level, but the Stoke City midfielder sliced his volley just over.
Snodgrass came on as Scotland went for broke with 10 minutes left and the midfielder snatched an equaliser in the 87th minute.
Fletcher lofted a pass beyond the Slovenia defence and Snodgrass beat Jan Oblak to it before hooking home.
Scotland poured forward in search of a priceless winner, but Oblak scrambled the ball away as Matt Phillips' deflected pass threatened to trickle over the line.
Cesar was shown a straight red card after feuding with Scotland's Christophe Berra just seconds after the pair were both booked.
But it was too late for Scotland to snatch a winner as Snodgrass's weak header wasted their last chance.