The hosts took the lead in the sixth minute of the contest through Simon Adingra, but an own goal from Adam Webster, which came five minutes after a red card for Dahoud, meant that the points were shared.
Brighton are eighth in the table, now boasting 19 points from their first 12 matches, while Sheffield United remain in 19th position, collecting five points from their opening 12 games.
Brighton had a chance to score in the fourth minute through Ansu Fati, but the on-loan Barcelona attacker sent his effort straight at Sheffield United goalkeeper Wes Foderingham.
The home side managed to make the breakthrough less than two minutes later, though, with Adingra sliding the ball past Foderingham after an excellent combination with Facundo Buonanotte.
Fati again broke through in the 15th minute, but Foderingham was on hand to make a smart save, with Sheffield United struggling to come to terms with Brighton's attacking movement early on.
Brighton had another chance in the 30th minute, with the ball falling kindly for Buonanotte inside the box, but the attacker missed the post in another let-off for Sheffield United.
The Seagulls came close to doubling their advantage just before the half-time interval, but Billy Gilmour's effort from distance was tipped over the crossbar by Foderingham.
Joao Pedro struck over the crossbar in the 51st minute, with the hosts dominating the contest, but Sheffield United were always in the match while the score remained just 1-0.
Brighton were reduced to 10 men in the 69th minute of the contest, with Dahoud given his marching orders for a heavy challenge on Ben Osborn.
The red card led to Carlos Baleba and Joel Veltman being introduced for the home side, but Sheffield United made the most of their man advantage in the 74th minute, levelling courtesy of an own goal from Webster, who turned a low cross from Jayden Bogle into the back of his own net.
The Blades then had a brilliant chance to take the lead shortly after the leveller, but Bogle fired wide of the post from 12 yards.
Brighton struggled to recover from the red card and were ultimately fortunate to collect a point, with Sheffield United finishing the stronger.
Brighton will be aiming to return to winning ways when they visit Nottingham Forest on November 25, while Sheffield United's league season will continue at home to Bournemouth on the same afternoon.