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.
Aaron Ramsey netted an injury-time winner to snatch a 2-1 victory for Arsenal over managerless Crystal Palace.
The south London club were rocked by Tony Pulis' departure by mutual consent on the eve of the season on Friday, but they performed admirably before taking a shock lead at the Emirates Stadium on Saturday.
Having comfortably absorbed waves of Arsenal pressure, debutant Brede Hangeland headed Palace and their caretaker boss Keith Millen into a 35th-minute lead.
James Chester's goal and a late Allan McGregor penalty save earned Hull City a 1-0 win and spoiled QPR's Premier League return.
Having dropped out of the top flight in 2012-13, Harry Redknapp's men bounced back at the first attempt last season courtesy of a 1-0 win in the Championship play-off final against Derby County.
The hosts were the better team in the first half at Loftus Road, but they failed to build on their good start as Steve Bruce's side clinched a narrow victory, McGregor denying Charlie Austin from the spot with five minutes to play.
Chester's goal arrived shortly after the interval, the centre-half proving too strong for QPR debutant Rio Ferdinand and glancing in Stephen Quinn's right-wing corner.
Laurent Koscielny levelled on the stroke of half-time, but the FA Cup holders looked like they would fail to capitalise, even as Palace winger Jason Puncheon departed for a second booking in the closing stages.
But Ramsey was on hand to prod home from point-blank range when Mathieu Debuchy's shot was saved, following a display of set-piece defending that would have left Pulis horrified.
Tottenham debutant Eric Dier struck in injury time to secure a 1-0 opening-day derby win at West Ham.
England Under-21 defender Dier, a close-season acquisition from Sporting Lisbon, found himself on the end of Harry Kane's measured throughball to round home goalkeeper Adrian and coolly convert.
A win looked unlikely for Tottenham, playing under new manager Mauricio Pochettino, when Kyle Naughton was sent off for handball in the 29th minute, although Mark Noble missed the resulting penalty.
It was 10-a-side when West Ham defender James Collins collected a second booking in the 63rd minute and both teams pressed in a frantic finale before Dier had the final word.
Chris Wood came off the bench to equalise with four minutes left as Leicester City drew 2-2 with Everton.
Playing their first top-flight match since 2004, Leicester left it late to deny Everton on Saturday, with Wood curling in from just inside the penalty area to delight the King Power Stadium crowd.
Everton, still reeling from the loss of Ross Barkley – who will miss six to eight weeks with a knee problem suffered on the eve of the campaign – started brightly and took the lead thanks to Aiden McGeady's stunning 20th-minute finish.
Record signing Leonardo Ulloa, who joined from Brighton and Hove Albion for a reported fee of £8million last month, provided a swift response within two minutes for Leicester, though.
With half-time looming, Everton took the lead again thanks to Steven Naismith's well-taken effort from inside the penalty area, but Wood's late strike ensured both sides would share the spoils.
Andreas Weimann scored the only goal as Aston Villa beat Stoke City to get their campaign off to a winning start.
Villa, who kicked off last season with a 3-1 win at Arsenal, had to defend well in the early stages, but looked increasingly threatening as the match opened up.
And Weimann found the target five minutes into the second half, sweeping home a right-footed effort from inside the penalty area amid some questionable Stoke defending to secure all three points.
New Stoke signings Bojan Krkic and Mame Biram Diouf failed to inspire a fightback for the home side at the Britannia Stadium, with Villa goalkeeper Brad Guzan enjoying a relatively quiet second half.
Sebastian Larsson's late leveller denied Alan Irvine three points on his West Brom debut as Sunderland snatched a 2-2 draw.
A brace from Saido Berahino cancelled out Lee Cattermole's thunderbolt opener for Sunderland and appeared to have handed Irvine a dream start as the new Premier League season got under way.
However, Larsson swept home a low strike in the 85th minute to ensure Gus Poyet's men left The Hawthorns with a point.