Holding the momentum from its inspiring run to the 2016 European Championship quarterfinals, Iceland topped their qualifying group and advanced automatically to the finals in Russia next year at the expense of favorite Croatia.
Iceland is the smallest nation in terms of population - 330,000 - to make the World Cup.
Sigurdsson put the hosts ahead in in the first half and Johann Gudmundsson made it 2-0 in the second.
Iceland won the group by two points from star-studded Croatia, which beat Ukraine 2-0 in Kiev.
Ukraine was another three points behind Croatia in third, Turkey fourth, and Finland fifth. Kosovo completed its maiden qualifying for a major tournament with a single point.
Qualifying was all the sweeter for Iceland because it fell at the last hurdle in 2014 World Cup qualifying - to Croatia.
In the tightest group of European qualifying, Iceland entered the last round with a two-point lead over Croatia and Ukraine.
The fans at packed Laugardalsvollur Stadium had to be patient but the wait was worth it.
Five minutes before halftime, Sigurdsson took the responsibility and scored. The attacking midfielder fired in his fourth qualifying goal following a neat solo effort in the penalty area.
Gudmundsson calmed all nerves when he met Sigurdsson's cross in the 68th minute, setting off a wild celebration.