Notching up a 100% record so far, the Swiss only need to avoid defeat against Belarus in order to make the next stage.
Switzerland got off to a dream start against Iceland as they took the lead as early as 58 seconds. Xherdan Shaqiri, goalscorer against Denmark, fed a neat ball through to Fabian Frei who took the ball to the keepers right and slotted it into an empty net.
Shaqiri was always lively and always wanting the ball. His trademark long shots were demonstrated as he rifled an effort at Haraldur Bjornsson, which the Icelandic goalkeeper managed to palm away.
At the other end, Gylfi Sigurdsson sent Switzerland a reminder that they were no pushovers by sending a powerful effort towards the Swiss goal, one of the reasons why Iceland managed to notch up 33 goals in only 10 qualifiers.
Xherdan Shaqiri was there to instigate a move again as he fed the ball to Frei, who found Innocent Emaghara, who was able to send the ball off the goalkeeper’s foot and into the net.
Iceland found themselves eliminated, despite giving a strong performance against the Swiss in which they limited the number of Swiss attacks.
Elsewhere in Group A, the hosts Denmark got back on track against Belarus.
After an opening day defeat at the hands of Switzerland, they put in an impressive display in order to beat Belarus and revive their hopes of making the semi-finals.
Nicolai Jorgensen scored a sublime effort. After missing out on a start against Switzerland, Jorgensen fully prooved that he is worth a starting place in this encounter.
After Dmitri Braga put Belarus ahead early on, Jorgensen won the penalty that would see Christen Eriksen convert the equaliser.
With 19 minutes left, the Bayern Leverkusen winger scored a long-range effort that rifled into the far corner to cement the Danes win.
Tonight, Spain take on Czech Republic, hoping to build on their opening day draw against England.
Stuart Pearce’s men face Ukraine and will be hoping to take one step further to qualification from the group.