The Scots' hero turned out to be centre forward Martin, who received some boos when he was brought on as a last-ditch hope for the home team but, within minutes, was being cheered to the rafters.
The victory breathed new life into the Scots' dismal campaign, putting them on to seven points in fourth place, now just one behind Slovenia, who slipped back to third in the table behind Slovakia, nine, and runaway leaders England on 13.
It also gave them fresh hope as they look forward to their next group match, a clash with their oldest rivals England at Hampden Park in June.
"The players can be proud of themselves. We have got ourselves back in contention and we look forward to the next game now," Strachan, whose own future as Scotland manager has been the subject of much speculation, told Sky Sports.
A half-empty Hampden had looked resigned to seeing Scotland fail to prevail against Slovenia in what Strachan had described as a "must-win" match.
The Scots were left cursing their bad luck after a first half during which Leigh Griffiths struck the woodwork twice in the space of a minute and Russell Martin had a headed goal ruled out for pushing.
Their chance of the crucial win looked lost when substitute Ikechi Anya missed a great chance but Chris Martin, brought on in the 81st minute, latched on to debutant Stuart Armstrong's threaded through ball to strike home a left-foot shot.
It was a remarkable finale, especially as Derby County striker Martin, who is on loan at Fulham, is not too popular with a section of the Scottish fans.
"Chris Martin gets a good reception from the players and the staff. That is why we pick him," Strachan said.
"He is in great company. Kenny Dalglish was booed, so was Gary McAllister, so was Alan Hansen. He must be some player."
Strachan also had extravagant praise for the Celtic midfield newcomer Armstrong. "I think his could be the best Scottish debut I have ever seen," said the manager.
(Reporting by Ian Chadband; Editing by Clare Fallon)
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