Giroud, who has played only 101 minutes in the English Premier League this season, converted a second-half spot-kick as injury-hit France stayed second in Group H behind Turkey who beat Albania 1-0, thanks to a last-gap Cenk Tosun goal.
France and Turkey, who meet at the Stade de France on Tuesday morning, have 18 points but the world champions are behind by virtue of their 2-0 loss in Konya in June.
Third-placed Iceland have 12 points, Albania nine, and Andorra are second to last on three points after beating Moldova 1-0 for their first-ever Euro qualifying points.
Andorra had lost all 56 of their previous Euro qualifiers, but Marc Vales ended that streak by scoring with a header in the 63rd minute, after Moldova had seen Radu Ginsari sent off for picking up a second yellow card.
France were without the injured Kylian Mbappe, Paul Pogba and Hugo Lloris and there was more bad news when N’Golo Kante picked up a muscle injury during the warm-up, leaving his place in the starting XI to Moussa Sissoko.
“It was a very tough game but we ground through and Olivier’s goal did the job,” said forward 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.”
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’ 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. – REUTERS