The Potters might well have taken an unlikely three points from the contest when they were awarded a penalty for a foul on substitute Jese in the closing moments at the bet365 Stadium, only for Charlie Adam to see his spot kick saved by Mathew Ryan.
The draw sees Stoke stay in the bottom three despite Paul Lambert keeping his unbeaten home record since taking over last month. The Staffordshire club could slip down into 19th place should Huddersfield Town beat Bournemouth on Sunday.
Both sides started the game with an impressive energy, but it was Brighton who conjured the first opportunity, as Jose Izquierdo profited from some careless defending before seeing his bending effort tipped over the bar by Jack Butland.
Albion were compact throughout the first half and that appeared to frustrate both a lacklustre Stoke and their fans, who started to make their collective feelings known with a smattering of jeers as the hosts failed to break down the Seagulls' defence.
Stoke were unable to produce much going forward and they soon found themselves trailing in the 32nd minute, when Izquierdo combined brilliantly with Jack Stephens in order to bend a curling finish beyond a helpless Butland.
Shaqiri had a quiet opening half, but the Swiss international might have made amends just before the break only to skew a volley of his own over Ryan's goal.
The Seagulls continued to look purposeful on the counter in the second period, this time as Stephens sent Brighton on the counter, leaving Pascal Gross to play in Davy Propper, but he headed over the bar.
Shaqiri was much improved after the break and he was somehow able to drag the Potters back into the contest, the playmaker curling a memorable finish into the bottom corner from the edge of the box to restore parity with Stoke's first shot on target after 68 minutes.
Brighton continued to push for a winner despite being frustrated by the Potters, and they might have found one through Propper, who perhaps should have taken a touch before sending Leonardo Ulloa's knockdown straight into the arms of Butland.
The best of the drama was still to follow, though, as Jese won a cheap penalty after a clumsy shirt pull from Stephens, but Adam could not punish the Seagulls as he saw his kick saved and rebound somehow kept out by a desperate Albion defence.
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.
STOKE CITY (4-5-1): Butland; Bauer, Shawcross, Zouma, Pieters; Fletcher (Berahino '46); Shaqiri, Allen, Badou (Adam '79), Choupo-Moting (Jese '62); Diouf
BRIGHTON & HOVE ALBION:(4-5-1) Ryan; Schelotto, Duffy, Dunk, Bong; March (Knockaert '86), Stephens, Propper, Izquierdo; Gross (Kayal '79); Murray (Ulloa '71)