After Xherdan Shaqiri's strike in the 68th minute had brought the Potters level, cancelling out Jose Izquierdo's 32nd-minute goal, the hosts had the chance to snatch all three points at the death when Bobby Madley judged Dale Stephens had fouled substitute Jese Rodriguez in the box and pointed to the spot.
But - with Jese looking annoyed not to be taking the penalty - Adam saw his effort from 12 yards kept out by Mathew Ryan. Anthony Knockaert then headed Mame Diouf's stoppage-time header off the line to ensure the contest ended all square.
Izquierdo demonstrated his threat as early as the second minute as his curling shot from just inside the box brought a fine save out of Jack Butland.
Stoke, without the injured Peter Crouch, made their first attempts on goal through Eric Maxim Choupo-Moting, who twice struck off-target, before Glenn Murray lifted an effort over Butland and the bar.
Shaqiri then hooked over on the turn inside the Brighton box, and Solly March tried his luck with a strike that was easily dealt with by Butland.
In the 28th minute Izquierdo had Stoke fretting again as he ran on to a great Dale Stephens pass, cut inside from the left and fired over the bar.
And four minutes later the Colombian started and finished a delightful move to put the visitors ahead, exchanging a one-two with March and then another with Stephens before slotting past Butland.It was another gem to add to his stunning effort in the win against West Ham last weekend.
Badou Ndiaye sent an effort wide moments later, but there was little thereafter from the home side in an attacking sense for the rest of the first half, while Davy Propper headed off-target for Brighton just before the interval.
Davy Propper headed over at the other end, and after Lambert then brought on Jese - for his first appearance since November - Shaqiri equalised, curling a shot from outside the area into the bottom corner.
Shaqiri subsequently appealed for a penalty after going down in the box following a tangle with Gaetan Bong, but referee Madley waved play on.
After Izquierdo had another shot saved by Butland, Stoke were presented with the golden opportunity to seal victory as Jese won a penalty in the 90th minute.
The Spaniard had to be calmed down by team-mates as he complained about not being given the chance to take the spot-kick - and then saw Adam's effort from the spot saved by Ryan.
There was then further frustration for the home side as Knockaert made a goal-line clearance to deny Diouf.
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