There were a lot of positive signs in the attacking third, with Jack Grealish and Iliman Ndiaye both excellent and Everton deserved the win in the end, but they had to survive several shaky moments with the Seagulls hitting the woodwork twice and having a penalty saved late on.
That penalty was a debatable one and represented the second week in a row that the Toffees had a spot kick given against them that was certainly very questionable.
And just as at Elland Road last Monday night, the Everton player concerned - on this occasion Kiernan Dewsbury-Hall - used his body to block the ball at very close range, and it struck his arm leading to the penalty being given.
But as with the James Tarkowski decision at Leeds, it is possible to see the situation either way and claim the penalty was legitimate or harsh, depending perhaps on which team you support.
However, these decisions are without a doubt creating the potential for a very messy set of circumstances in future Premier League matches.
The continuing confusion over what is and is not a penalty and the endless VAR uncertainties is going to rumble on and surely cause further controversy over and over again in the coming months.
As for Everton, one other take from the Brighton win is the fact that the Toffees again looked unusually vulnerable defensively at times.
This was an issue on Monday evening too, and obviously can largely be explained by the continuing absence of Jarrad Branthwaite and Vitalli Mykolenko.
David Moyes continued to use James Garner at left-back with Jake O'Brien taking the right-back position as he so often has recently since midway through last season.
While it is clearly a short-term emergency solution and will not continue throughout the campaign, it does again highlight the Blues transfer struggles early on in the window to find specialist full-backs.
One other issue that is a possible cause of concern is Tarkowski's regular lapses in the heart of the defence. He made several poor decisions, including one casual back-pass that almost ended in a goal.
Is it just a few bad moments that will disappear soon enough, or is the players' age and the wear and tear of so many hard Premier League campaigns beginning to catch up on him?
With Branthwaite out for an unknown amount of time, Moyes will be hoping that his captain is able to recapture that previous consistently reliable form that made him such a solid stopper.
Hopefully, Everton will have enough time in the transfer window to add the two or three additional players that the squad still needs, and recent rumours suggest a new right-back could be near to arriving.
The other area where the Blues also really do need fresh blood is central midfield, where Idrissa Gueye had a few wobbly moments early on before improving later in the game.
He needs support in this role and another player here would also bolster the defensive solidity of the side, especially while there are so many injuries and new signings bedding in.
Individuals are out there who could solve this problem, and stories are circulating that Tottenham's Yves Bissouma is now very much on the Toffees' radar.
All in all, then, a great way to start life at the stunning, state-of-the-art new home of the club, and it also seems certain that Everton will have a very busy last few days in the summer transfer window.