The Swansea City midfielder converted a penalty and fired home a superb volley in the first half of Monday's Group A clash in Reykjavik as Guus Hiddink's men fell to a third defeat in four games since he retook the reins.
Iceland's win maintained their place as pool leaders on goal difference having begun the campaign with 3-0 wins over Turkey and Latvia.
Borek Dockal, David Lafata, Ladislav Krejci and Tomas Necid found the net to render Yuriy Logvinenko's late double irrelevant as the Czechs kept up their 100 per cent record.
In the group's other game, Valerijs Sabala's penalty earned Latvia a home 1-1 draw after Bilal Kisa had struck for Turkey.
Radja Nainggolan's 51st-minute goal secured Belgium a 1-1 Group B draw at Bosnia-Herzegovina.
Edin Dzeko's powerful finish put the home side in front before Nainggolan's effort slid under goalkeeper Asmir Begovic as the spoils were shared.
But Vincent Laban halved the deficit from a set-piece before Andy King's red card ensured a nervous finish.
Omer Damari was the star, bagging a hat-trick that was added to by Tomer Hemed's late penalty.
Graziano Pelle netted on his international debut as Italy defeated Malta 1-0 in Ta' Qali in a game that saw both sides reduced to 10 men.
Michael Mifsud was sent on off soon after, but Italy could not capitalise. Antonio Conte's side struck the woodwork four times and saw Leonardo Bonucci dismissed late on for a professional foul.
Italy's win keeps them level on points with Croatia, who hammered Azerbaijan 6-0 as Ivan Perisic's double was supplemented by efforts from Andrej Kramaric, Marcelo Brozovic, Luka Modric and a Rashad Sadygov own goal.
A place further below are Norway, who beat Buglaria 2-1 in a game which saw Martin Odegaard become the youngest player to appear in a Euro qualifier at 15 years and 300 days old.
But Havard Nielsen was the hero, his 72nd-minute winner coming after Nikolay Bodurov had cancelled out Tarik Elyounoussi's opener.
