Graziano Pelle had given Feyenoord the lead after just six minutes, but Sigthorsson equalised from the penalty spot 14 minutes before the break.
And the Icelandic striker netted his second just six minutes later to hand the champions all three points and leave Ronald Koeman's men pointless after their opening three matches.
FC Twente maintained their unbeaten start to the season as a stunning first-half performance helped them to a 6-0 demolition of Utrecht.
Quincy Promes put them ahead after eight minutes and three minutes later the lead was doubled as Dusan Tadic netted from the penalty spot.
Luc Castaignos scored in the 25th and 36th minutes to make it 4-0 at half time and Promes grabbed his second after the break before Rasmus Bengtsson completed the rout in the last minute.
Heerenveen lost their first match of the new season as they were beaten 4-2 at home by Heracles Almelo.
Alfred Finnbogason had the hosts ahead but Lerin Duarte equalised on the stroke of half-time and two goals in seven second-half minutes from Ben Rienstra and Mike te Wierik put Heracles in control.
Joey van den Berg grabbed a goal back for Heerenveen in the 73rd minute but Bryan Linssen's strike with two minutes remaining secured victory for the visitors, who picked up their first points of the season.
A Michiel Kramer goal in time added on saw ADO Den Haag get off the mark with a 2-1 win at NAC Breda.
Aaron Meijers put the visitors ahead in the 76th minute but Alex Schalk's penalty with five minutes remaining looked to have secured a draw before Kramer's late show.