The Welsh side took the lead by the 22nd minute through Sam Clucas and could have been up 2-0 before the break if Jordan Ayew’s goalmouth effort had gone successful.
The 26-year-old robbed Per Mertesacker of the ball in his own area but instead of directing his effort beyond onrushing Petr Cech or squaring for an unmarked Abraham for a tap-in, the Ghanaian went in for a chip which was easily saved.
Two quick goals from Sead Kolasinac and Aaron Ramsey in the second half turned the game on its head for Arsene Wenger to win his 800th game in charge of the Gunners.
“It [Ayew's chance] was a big chance and 2-0 at half-time would have made it a totally different situation, but those are the fine margins we are dealing with," Lambert said.
“If that ball had gone in, it could have been different [and we would've won the game].
“The first goal we conceded was fortunate for them because the ball had bounced off a few of our defenders before fallen to their wing-back.
“I am more disappointed with the second goal. We did not get the details right in that, we were too open and made some mistakes.
“I think we were worth a point, we played well in the first half. You always know it is going to be a tough task coming here against a strong attacking side with quick combinations in and around the box."
He added: “We knew we would need to be focused for the whole game and in the first half we played well and took our chance on the counter-attack, which is where we thought we would get our opportunities.
“It was just that period after half-time where we did not get our details right, we did that so well in the first half."
Ayew lasted 81 minutes before being replaced by Luciano Narsingh.
Swansea sit 17th position with eight points and will welcome Brighton and Hove Albion next weekend before a trip to Burnley.