Ten Hag gave new signings, Lisandro Martinez and Chrisian Eriksen their official debuts while Marcus Rashford led the line ahead of Cristiano Ronaldo, who started from the bench.
Man United threatened in the opening exchanges as Bruno Fernandes had a chance in the box after a cut-back by Scott McTominay but he blasted his shot over the crossbar.
Brighton drew first blood on the half-hour mark after Danny Welbeck went through on goal on the left channel and played a cut-back that was slotted into a gaping net by Pascal Gross, who was unmarked at the back post.
Graham Potter's charges gained the momentum and Gross doubled their lead just before the halftime break with another strike at the back post from a rebound after goalkeeper David de Gea parried Solomon March's shot on his path.
Ten Hag eventually brought on Ronaldo and sacrificed Fred in a tactical change early in the second stanza and United improved going forward.
Rashford should have reduced the deficit for the Red Devils on the hour-mark after Ronaldo's through ball found him unmarked in the box from a counter-attack but his side-footed shot was unbelievably saved by goalkeeper Robert Sanchez in a one-on-one.
However, United maintained their pressure and eventually got a goal back 20 minutes before time after a scramble in the box from a corner bounced off Diogo Dalot and Alexis Mac Allister could not keep the ball out of danger near the goal-line and scored into his own net.
The Red Devils worked their way up field as 12 minutes before time Donny van de Beek replaced McTominay for attacking impetus in midfield.
However, the Seagulls were able to slow down United's attacks from their own substitutions and were disciplined at the back as they held on to claim a famous win and a perfect start to the league season.