Hansen's goal now makes it almost impossible for Scotland to claim the one automatic qualifying spot and, on this evidence, they will be struggling even to take second place in Group Nine behind the runaway leaders, the Czech Republic.
Allan Johnston's first goal in his fifth international appearance gave Scotland the edge. But Craig Brown's team were left facing an uphill task after Matt Elliott became only the 11th player in the nation's history to be sent off when he was caught lashing out at striker Todi Jonsson. It was an unnecessary act of violence by the Leicester City defender, who now automatically misses the next qualifier against the Czech Republic on Wednesday. His actions in this, his seventh international appearance, must now raise a question mark over his future at this level. He had been recalled to the side only because Colin Hendry was sidelined with a groin injury.
Scotland's coach, Craig Brown, said: "Elliott's future is in some jeopardy. You've got to have players you can rely on. I know he was under provocation, but he is experienced enough to know he should have been able to handle it. There was a case that Jonsson may have feigned injury and, as far as I'm aware, no contact was made. However, Elliott lifted his hand and that is totally unacceptable in my book. I told him that at half-time and he apologised profusely. He was very distraught and very disconsolate."
The Blackburn striker Kevin Gallacher was also recalled, in the absence of Don Hutchison - who scored the winner against Germany - because the Everton striker was getting married.
Scotland narrowly defeated the Faroes 2-1 last October in Aberdeen and Brown had been looking for an improvement on that performance yesterday evening. But in this, his 50th match in charge, Scotland were to regret squandering chances and that moment of madness from Elliott.
The Faroes went into the game with only one point in the group behind them after five qualifiers but they refused to be overawed.
Scotland, however, should have sewn the game up by half time. Kilmarnock's Ian Durrant whipped in a corner from the right in the 12th minute and Elliott's header was superbly tipped over by keeper Jakup Mikkelsen. Scotland had a scare two minutes later when Todi Jonsson ran clear but Sullivan was able to smother his shot.
Scotland stepped up the pressure and first Elliott and then Billy Dodds saw their efforts turned away by the Faroes keeper. Gallacher then miscued from a good position after David Weir had flicked on a corner from Durrant.
Johnston made the breakthrough seven minutes before the interval when Mikkelsen could only parry a driven cross from Gallacher.
However, Scotland were reduced to 10 men right on half-time when Elliott received the red card after he and Todi Jonsson had exchanged words on the edge of the penalty area.
Brown rethought his tactics at half-time, bringing on Colin Cameron for Durrant, who had worked tirelessly in the first half. The Scots were looking comfortable and Callum Davidson almost added a second when Dodds cut the ball back but Mikkelsen managed to gather safely. Scotland were playing out time, only to cave in during stoppage time when Hansen scored that dramatic equaliser.