Le Sommer struck from the spot after Wendie Renard's own goal had cancelled out Valerie Gauvin's opener for France at the start of the second half.
The victory put Corinne Diacre's team top of Group A with a maximum six points.
Norway, the 1995 champions, are second on three points, ahead of Nigeria on goal difference after the African champions won 2-0 against South Korea, who are bottom with no points.
Norway play South Korea in their final encounter while France need a draw against Nigeria in their last group game to advance as group winners.
Norway were on the back foot early on before stepping up a gear after 10 minutes, threatening on the break while the French defence showed signs of nerves.
Norway had a clear chance in the 14th minute when Ingrid Engen's header from a corner was deflected over the bar by Amel Majri.
Les Bleues enjoyed more possession and were rewarded for their efforts a minute into the second half when Gauvin, who had started the opening game on the bench, tapped home from Majri's cross.
Yet eight minutes later, Renard deflected Isabell Herlovsen's cross into her own net.
It was all forgotten in the 72nd minute, however, as France were awarded a penalty following a VAR review after Marion Torrent was kicked in the area by Engen.
Le Sommer coolly converted the spot kick to send the Allianz Riviera stadium into raptures, netting her fifth World Cup goal to equal Marie-Laure Delie as France's all-time top scorer in the tournament.
(Writing by Julien Pretot; Editing by Toby Davis)
