Everton started the first half strongly and thought they had taken the lead after just six minutes when Jack Harrison scored from a corner, but the goal was immediately ruled offside by the linesman, and a swift VAR check confirmed the decision.
The visitors grew into the game following that chance, and Joao Pedro came close to scoring their first goal of the 2024-25 season as the forward struck the post with a fizzing effort in the 10th minute.
Both teams had several chances in the subsequent 15 minutes, but it was Hurzeler's side that broke the deadlock in the 26th minute when Yankuba Minteh raced down the right wing and cut the ball across the box for Kaoru Mitoma to tap into the empty net.
The hosts may have felt hard done by going into the break 1-0 down after a strong first 45 minutes, but they believed they had a chance to equalise right after the restart when a penalty was awarded, only for VAR to intervene and the referee overturned the decision.
Brighton dominated the remainder of the match, finding the second goal in the 56th minute when Danny Welbeck calmly finished into the bottom corner after capitalising on an Everton mistake.
Ashley Young added to the misery as he was given his marching orders in the 66th minute after dragging Mitoma to the floor with the winger racing through on goal.
Brighton threatened to add to their goal tally in the closing stages of the match, and Simon Adingra eventually added a third in the final minutes to top off an excellent outing in Hurzeler's first competitive match in charge.
EVERTON VS. BRIGHTON & HOVE ALBION HIGHLIGHTS
26th min: Everton 0-1 Brighton (Kaoru Mitoma)
Mitoma started the move by charging through the midfield before the ball was eventually shifted to the right, where Minteh breezed past Vitaliy Mykolenko and played a low ball across the box, with Mitoma sweeping home at the back post.
56th min: Everton 0-2 Brighton (Danny Welbeck)
Idrissa Gueye lost the ball after a poor pass, with Mats Wieffer intercepting and playing into the path of the onrushing Welbeck, who feinted right and cut onto his left, placing his shot into the bottom corner to give Brighton the second goal.
66th min: Everton 0-2 Brighton (Ashley Young Red Card)
A long diagonal ball was played towards Mitoma and Young was able to cut it out, but a poor touch from the defender allowed the Brighton forward to intercept and race towards goal, with Young pulling him to the floor and being given his marching orders.
87th min: Everton 0-3 Brighton (Simon Adingra)
Adingra put the cherry on top in the final minutes of the game as the winger was fed the ball by Welbeck, cutting across the box and slotting into the bottom left corner with his weaker foot.
MAN OF THE MATCH - DANNY WELBECK
The veteran striker has seen plenty of attacking options come through the door at Brighton in recent seasons, but Welbeck has deservedly kept his place in the first team, and he showed why today.
Welbeck expertly ran the channels and linked up most of Brighton's positive attacking moves, including the pass out to Minteh before the first goal, scoring the second and finding Adingra with a beautiful outside-the-foot pass for the third and final Brighton goal.
The 33-year-old's 56th-minute goal means the striker has scored a goal in the Premier League in all of the last 15 seasons, a brilliant record that sums up Welbeck's consistency.
EVERTON VS. BRIGHTON & HOVE ALBION MATCH STATS
Possession: Everton 38-62 Brighton
Shots: Everton 9-10 Brighton
Shots on target: Everton 1-5 Brighton
Corners: Everton 1-5 Brighton
Fouls: Everton 8-8 Brighton
BEST STATS
WHAT NEXT?
A disappointing 3-0 defeat for Everton will be followed up by a difficult trip to face Tottenham Hotspur next Saturday, a side they have failed to beat in any of the last seven meetings.
Brighton, on the other hand, will be delighted with a strong performance and comfortable victory in their first outing under Hurzeler, and they will be hoping to follow that up against Manchester United at Old Trafford in their next outing.