Haaland, City’s headline summer signing, netted both goals as the champions saw off West Ham 2-0 at the London Stadium while United, with Cristiano Ronaldo among the substitutes, toiled in a 2-1 home loss to Brighton.
There was also drama in the day’s other match as Brentford fought back from 2-0 down to claim a 2-2 draw at Leicester.
Haaland was quick to make his mark for City, opening the scoring with a penalty 36 minutes into his Premier League debut.
The Norwegian himself won the spot-kick after being brought down by Hammers substitute goalkeeper Alphonse Areola.
Areola, who had replaced the injured Lukasz Fabianski after 29 minutes, was beaten again in the second half as Haaland raced onto a Kevin De Bruyne through-ball to tuck home.
United’s hopes of getting off to a winning start under Ten Hag were dashed after two first-half goals from Pascal Gross at Old Trafford.
The home side were outplayed for large spells by the Seagulls, who had not won in nine previous league visits to the ground.
Ten Hag turned to Ronaldo, who has indicated a desire to leave the club, after the break and his side did hit back when Alexis Mac Allister turned into his own net after 68 minutes.
Yet despite applying late pressure, the hosts could not find a way through and Brighton held on to secure back-to-back wins over United after their 4-0 success at the Amex Stadium in May.
Substitute Josh Dasilva scored Brentford’s equaliser with four minutes remaining in a pulsating encounter at the King Power Stadium.
Timothy Castagne headed the Foxes into a 33rd-minute lead and Youri Tielemans hit the post before Kiernan Dewsbury-Hall doubled the lead in the opening minute of the second half.
Ivan Toney gave the Bees hope just after the hour and, although the Foxes went close again when Wesley Fofana hit the woodwork, Dasilva had the final word with a stunning strike.