Incredibly, substitute Anthony Knockaert then came to Brighton's rescue by clearing Mame Biram Diouf's header off the line from Adam's corner in a frantic ending.
The draw still leaves Stoke in the midst of a potential relegation scrap, while Chris Hughton's men remain within three points of the dropzone despite sitting in 13th place in the standings. It was a superb team goal, with Solly March and Dale Stephens involved.
At half time Stoke withdrew Darren Fletcher, to ironic cheers from the home fans, for Saido Berahino and tweaked their shape.
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With the home support roaring their team on, Stoke continued to push forward for the victor and, when Bobby Madly pointed to the spot in stoppage time, it looked like they'd found it. Brighton failed to get close enough to the Switzerland winger, whose precise effort found the bottom-left corner with Stoke's first shot on target.
Then came the most controversial moment of the match as a fairly weak challenge from Stephens on Jesé Rodriguez Ruiz was deemed a penalty by referee Bobby Madley. That was the precursor to a stunning ending that started when Jese was felled by Stephens.
However, the ball rebounded back into the path of Adam, only for the sluggish 32-year-old to be beaten to the the ball by a quick-thinking Lewis Dunk, who tackled the sloth-like Adam and cleared the danger.
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As Charlie Adam, who had only been on the pitch 11 minutes, stepped up and sidefooted the ball towards Ryan's right hand side, the Australian was at full stretch to make a brilliant save.
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