The Nordic side were trailing to two Ahmed Musa goals when they were handed a lifeline from the spot but Sigurdsson scooped his penalty kick over the bar and the Nigerians ran out 2-0 winners.
Iceland had started the brighter of the two sides in a first half that failed to produce a shot in target from Nigeria.
That changed early in the second half, however, when Musa brought a ball into the box down impressively and smashed in from close range.
And after he curled a shot off the crossbar, Musa blazed down the left flank, cut inside and rounded the keeper to double the leader.
A foul in the box awarded following video review with 10 minutes left when Finnbogason was felled handed Sigurdsson a great chance from 12 yards but although he sent the 'keeper the wrong way, he couldn't keep the shot down.
"It's tough to take," Sigurdsson was quoted by Goal.com as saying afterwards. "It's probably the toughest moment of my career. Hopefully I can put it right and score in the next game.
"It was a game of two halves. We were good in the first half and very organised defensively. I don't think they had any shots in the first half. We created numerous chances in the first half that we probably should have scored from.
"In the second half we were unlike ourselves. We were very open, especially after the first goal and we had to go and get something out of the game. Overall it's very disappointing.
"We were impatient in trying to win the game instead of letting it happen naturally. We're very disappointed because we felt after the first half the game was there for the taking.
"We're just going to have to get something out of the next game to get through."