Giroud broke the deadlock from the spot in the 65th minute, and France held on to seal a vital three points.
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France remain second on 18 points, the same as leaders Turkey who needed a last-gasp Cenk Tosun goal to beat Albania 1-0. Iceland are now six points adrift of top in third.
France and Turkey, who meet at the Stade de France on Monday, but the world champions are behind by virtue of their 2-0 loss in Konya in June. Andorra are second to last on three points after beating Moldova 1-0 for their first Euro qualifying points.
"It was a very tough game but we ground through and Olivier's goal did the job," said Antoine Griezmann.
"We had several chances to add to our goal and it would have been nice because they are dangerous on set pieces," said coach Didier Deschamps.
"But it's a good victory ahead of Monday's clash."
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France were without the injured Kylian Mbappe, Paul Pogba and Hugo Lloris and there was more bad news when Ngolo Kante picked up a muscle injury during the warm-up, leaving his place in the starting 11 to Moussa Sissoko.
Both teams struggled to create chances in a poor first half as Les Bleus failed to supply Griezmann, who had the best opportunity with a blocked shot on the turn three minutes from the break.
The game turned France's way when Griezmann was brought down in the area by Ari Skulason, with Giroud sending Hannes Halldorsson the wrong way from the spot to score in the 66th minute.
With 37 international goals, Giroud is now four goals shy of Michel Platini's tally of 41, with only Thierry Henry having scored more for France (51).
France came close to doubling the tally when Blaise Matuidi's low shot hit the post but Deschamps's team remained under threat until the final whistle.
They could have ended the night alone on top of Group H but Turkey prevailed against Albania when Tosun found the back of the net in the final minute with a close-range effort.
Additional reporting from Reuters