Van Gaal was taking charge of his first competitive game as United manager, but his new side exhibited many of the same failings from last season -- when they finished seventh -- as they slumped to a surprise defeat.
South Korea midfielder Ki Sung-yueng exploited inattentive marking to give Swansea a 28th-minute lead with a left-foot shot from the edge of the box.
United's new captain Wayne Rooney equalised early in the second period, but Gylfi Sigurdsson condemned the hosts to a first opening-day home defeat since 1972 with a close-range finish in the 72nd minute.
"It is very disappointing that we have lost our first home match," said Van Gaal, who took over at United after leading his native Holland to the World Cup semi-finals.
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"We were very nervous in the first half, made the wrong choices, and that is a pity. In the second half, we never played as a team. Therefore, I'm responsible."
Injuries obliged Van Gaal to give competitive debuts to Tyler Blackett, 20, and 21-year-old Jesse Lingard, who took their places in the 3-4-1-2 formation that had served United well in their unbeaten pre-season campaign.
United bossed possession in the first half, but Swansea struck from their first sight of goal, with Ki stroking home a lay-off from Sigurdsson.