Everton star Gylfi Sigurdsson got Iceland's opener in a 2-0 home win that makes the tiny country with a population of 335,000 the smallest nation ever to progress to a World Cup.
England's conquerors from Euro 2016 are the only country to qualify for a World Cup with a population of fewer than one million.
Sigurdsson scored just before half-time then turned provider for Gudmundsson to ensure victory with a close-range effort.
Iceland coach Heimir Hallgrimsson, a part-time dentist, said: "The hardest thing was to get going again after our massive party in France.
"And what's more we are in a group contested between Croatia, Turkey, Ukraine and Finland, who are much better than their position in the group would have you believe.
"The success is not an end in itself but a long journey towards a final destination."