Swansea were, in the most part, hard to break down, risky when pressing forward, especially down the right flank where Modou Barrow ran Nacho Monreal ragged, and after the sending off to Granit Xhaka, were certainly the team in the ascendency.
"That was an exciting game", Bradley said, "but it didn't finish the way we wanted". But soon Bob Bradley's boys made it 3-2 as Borja Baston scored.
Arsenal boss Arsene Wenger has confirmed that the club will not be appealing Granit Xhaka's recent red card.
There was precious little for Swansea at the other end, though the in-form Leroy Fer - who has four goals already this season - sent a rising drive just over the crossbar.
Xhaka was sent off for a cynical foul from behind on Modou Barrow, a red card with which Wenger did not have too many complaints, and against which he will not appeal.
Everton went ahead in the 64th through Romelu Lukaku, his sixth goal in the last five league games.
Arsenal's number 11 has been at the peak of his career in the past two seasons and as he turns 28, he will be hope to make the most of his peak years and help Arsenal to finally win a major trophy.
He said: "He could have had a hat-trick, maybe even four".
While Swansea strive to add to their solitary win at Burnley back in August, Arsenal have now won their last six in the league and sit level on points with Manchester City at the top of the table.
"When we were down to 10 men at 3-2, we could have scored the fourth goal but we could have conceded as well".
"It's always hard after the worldwide break for everyone to get back into a rhythm, but I would rather have had a scruffy game and win so that's all that matters", he said on the BBC's Final Score programme.
Arsenal made its early territorial dominance pay in the 26th minute as Sanchez's looping pass found Hector Bellerin on the right of the box.
Ozil restored the two-goal edge just before the hour mark.
The game could have been, maybe should have been one of the most one sided games for Arsenal, but in the end it was nerve wracking.