Sigurdsson, back at Swansea after signing from Tottenham Hotspur, fired in from close range to stun the hosts who had drawn level through skipper Wayne Rooney shortly after halftime.
Hopes were high for a winning start to the new season for the 20-times English champions after last season's woes, but the problems they suffered under David Moyes were evident again.
United had plenty of possession in the first half but lacked invention and pace and Swansea took the lead with their first real attack when Ki Sung-yueng slotted home after 28 minutes.
Rooney levelled with an overhead flick after 53 minutes and struck the post with a curling free kick but Swansea took advantage of some poor defending by United left back Ashley Young to pocket three unlikely points.
United rarely looked like equalising with fans drifting away from Old Trafford long before the end.