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.
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.
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.
