The Iceland international struck late on to give Garry Monk's men a famous victory, despite Wayne Rooney cancelling out Ki Sung-Yueng's first-half opener with an acrobatic strike
Sigurdsson bagged the winner 18 minutes from time on Saturday as Van Gaal - tasked with improving the fortunes at Old Trafford after David Moyes' disastrous spell in charge last term - endured a day he will want to forget in his opening Premier League fixture.
The Dutchman's 3-4-1-2 system did not appear to suit United despite being used in pre-season and they found themselves behind when Ki Sung-Yueng scored from 20 yards just before the half-hour mark.
Van Gaal, known for his tinkering, swapped to 4-4-1-1 in the second half and it appeared to be the tonic for United's response, led by new captain Wayne Rooney, who levelled with a clever overhead kick eight minutes after the break.
Rooney saw an outstanding free-kick crash against the post with 24 minutes left and Old Trafford was then stunned when Sigurdsson - Swansea's main signing of the close-season - finished well after Wayne Routledge's cut-back.
The result is sure to put early pressure on Van Gaal and is likely to see further spending from United in the transfer window, as they look to recover from last term's seventh-place finish.
The victory was Swansea's second at Old Trafford in 2014 - and second ever - as the Welsh club also won a third-round FA Cup tie by the same scoreline in January.
Van Gaal gave 20-year-old defender Tyler Blackett a Premier League debut, while Spanish midfielder Ander Herrera and Jesse Lingard also made their top-flight bows.
Lingard started in the unfamiliar position of right wing-back and was nearly caught out just six minutes in, when Routledge drifted across him untracked, but the Swansea man's first touch meant that Jonjo Shelvey's ball in behind was squandered.
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Sigurdsson - back at Swansea for a second spell, this time permanently, after signing from Tottenham - saw his 25-yard shot in the 12th minute fall awkwardly for United goalkeeper David de Gea, but the Spaniard did well to smother it.
Lingard was then injured in a challenge with Ashley Williams, and replaced by Adnan Januzaj, before the hosts eventually got going, with Lukasz Fabianski - the only debutant for the visitors from the start - needing to be alert to keep out Phil Jones' deflected cross and Juan Mata's inswinging effort.
Swansea were able to convert one of their chances, though, as Ki delighted the travelling fans in the 28th minute.
The South Korea international was played in by Sigurdsson, who initially linked with Nathan Dyer, and he hit a sweet left-footed effort that curled into the right corner from 20 yards.
Van Gaal's men failed to mount a significant response before the break and as a result, he replaced the ineffective Javier Hernandez with Nani and changed systems.
Rooney then produced the equaliser eight minutes into the second half, showing great anticipation to thrash Jones' near-post flick-on home from four yards in sensational style.
The England star continued to threaten, with Williams required to deny him with a last-gasp tackle, before he was then thwarted by the woodwork as his wicked 25-yard set-piece left Fabianski rooted to the spot, but crashed back off the post.
And Swansea made the hosts pay for their profligacy as a quick free-kick, taken by Wilfried Bony, set them free.
A cross from substitute Jefferson Montero made its way through to Routledge at the far post, and although he slipped, his ball fortunately fell for Sigurdsson, who hammered an effort through a sea of bodies that was too hot to handle for De Gea.
United, so often saved by late goals, rarely looked like levelling, but were denied a penalty despite Januzaj's cross striking the arm of Angel Rangel in injury time, as Swansea claimed an excellent win against the odds.