Stankovic was taken to hospital at half-time with a concussion after struggling in the latter stages of the first half.
"He suffered a knock midway through the first half and felt dizzy in the dressing room, but a thorough scan showed that he is alright," Serbian Football Association spokesman Aleksandar Boskovic told a post-match news conference.
"We are looking forward to his speedy recovery so we can have him back for the last two qualifiers at home to Italy and away to Slovenia," added coach Vladimir Petrovic.
The result lifted the Serbs to second place in Group C, behind winners Italy who secured a berth in next year's finals in Poland and Ukraine with a 1-0 home win over Slovenia.
Italy have 22 points from eight matches, ahead of Serbia on 14 and Estonia, who have played a game more, on 13. Fourth-placed Slovenia have 11 points from nine games and only a slim chance of clinching the runners-up spot.
Serbia are in the driving set to finish second as they need one win from their remaining two matches, at home to Italy on October 7 or away to Slovenia four days later.
The winners of Europe's nine qualifying groups and the best second-placed team will qualify automatically for the finals while the other eight runners-up will enter a two-leg play-off for the remaining four berths.
The Serbs, who had revitalised their hopes with Friday's 1-0 win over Northern Ireland in Belfast, launched an early onslaught against the Faroes which paid immediate dividends.
Winger Milan Jovanovic put Serbia ahead in the sixth minute with a looping header after a deflected cross from the left by Aleksandar Kolarov, who turned provided for a spectacular second midway through the first half.
The left-back set up Zoran Tosic with a well weighted pass and he volleyed home an unstoppable shot from the edge of the penalty box.
The visitors pulled one back against the run of play in the 37th minute after Frodi Benjaminsen took a searching diagonal pass into his stride and steered the ball past advancing keeper Bojan Jorgacevic.
Jovanovic missed two sitters and Tosic had a stinging shot palmed away by the keeper before central midfielder Zdravko Kuzmanovic made it 3-1 with a scorcher from 20 metres in the 69th minute.
Tosic and Kolarov hit the woodwork in the closing stages as Serbia enjoyed most of the possession in front of 10,000 fans at Partizan's stadium.
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