The Nordic nation have never qualified for a major tournament but only missed out on reaching the 2014 World Cup in the play-offs against Croatia.
Turkey goalkeeper Onur Kivrak struggled throughout Tuesday's match and it was his error that allowed Jon Dadi Bodvarsson to head Iceland into a deserved 18th-minute lead.
And after Burak Yilmaz spurned a great chance to equalise for Turkey, Kivrak tipped Gylfi Sigurdsson's 25-yard effort onto the post but saw the ball bounce and spin over the line.
Ajax striker Kolbeinn Sigthorsson wrapped up the win with a 77th-minute strike as Iceland – who have risen from 131st to 46th in the FIFA rankings over the last two-and-a-half years – made an excellent start to their qualifying campaign.
But despite being in action just six days ago, it was Iceland – who last played in June – who looked far more threatening in the opening stages.
Birkir Bjarnason and Sigthorsson had early half-chances before Ireland were denied by the woodwork in the 14th minute.
Bodvarsson met Ari Freyr Skulason's cross with power but the effort came crashing back off the crossbar with Kivrak beaten.
Kivrak looked decidedly shaky through the first half and the Trabzonspor goalkeeper was to blame for the opener as he spilled a routine attempt to catch Sigurdsson's corner.
Turkey's first chance came when a poor clearance found Selcuk Inan 25 yards out, but he could only curl over, before Bjarnson headed into the side netting at the other end with eight minutes of the first half to play.
Iceland's cause was made easier just before the hour-mark when Toprak, marking Sigthorsson, handled a long ball and was deservedly sent off.
And they were made to pay as although Sigurdsson's left-footed effort was initially clawed onto the woodwork, the spinning ball had enough on it to screw back over the line.
