The Iceland international midfielder, who had a loan spell with the Welsh club from Hoffenheim in 2012, was transferred to the Swans on Thursday as part of the deal which took Ben Davies and Michel Vorm to Tottenham.
In front of nearly 5,000 at Home Park on Sunday, the 24-year-old helped set up a goal for Jordi Amat and then netted twice himself having entered the friendly fray as a second-half substitute.
Northern Ireland Under-21 striker Rory Donnelly started up front in a Swans side that also included new signing Lukasz Fabianski in goal, while young midfielder Josh Sheehan was named in the line-up too.
Donnelly opened the scoring in the ninth minute when he pounced on a loose ball and saw his effort trickle over the line to the despair of Argyle goalkeeper Luke McCormick.
Chico Flores limped off with what looked like a back injury just before the break, with Ashley Williams replacing him.
Ecuador winger Jefferson Montero, another of this week's new signings, was one of several half-time changes and after another flurry of substitutions midway through the second period, Swansea scored again.
Sigurdsson swung in a 67th-minute corner and the rebound off Curtis Nelson fell to Spanish centre-back Amat, who stabbed home.
Midfielder Alex Bray had come on alongside Sigurdsson, and the duo combined in the 72nd minute for a third goal. Bray cut the ball back and the former Spurs man was on hand to slot in a neat finish.
There was still time for Sigurdsson to wrap up a convincing victory with a 25-yard free-kick in stoppage time.
Swansea stay in the South-West for a clash at Exeter on Tuesday night, while Argyle host Yeovil on Wednesday.