Central defender Barnett, snapped up from Luton Town during the summer, capped a stirring fightback with a 73rd minute strike.
Albion had been trailing from the 27th minute after Ryan Shawcross's third goal of the season had given a rugged Stoke side, who are unbeaten in five games, the lead.
Stoke have won 14 of the last 22 league meetings between the sides with Albion having just one victory to their name during that sequence of results.
Stoke had looked set to make it win number 15 when on-loan Manchester United central defender Shawcross celebrated his 20th birthday 24 hours early.
But Albion refused to buckle and were rewarded when Barnett reacted fastest in a goalmouth scramble to bundle home a left wing free-kick from Filipe Teixeira to score his second goal for the club.
Albion may have surrendered their 100 per cent home record but a share of the spoils was the least they deserved after dominating the game for long spells.
Albion must have wondered how they went into half-time trailing because, apart from a shot from John Eustace and the goal from Shawcross, Stoke rarely threatened.
Robert Koren ran the show for Albion setting up a chance for Ishmael Miller that was saved by Steve Simonsen who also denied Teixeira with a smart stop.
But Albion found Stoke a tough nut to crack and were made to pay in the 27th minute.
Stoke were a constant threat in the air and it was no surprise that was the way they made their breakthrough.
When Liam Lawrence lifted in a left-wing cross after collecting a misplaced shot by Dominic Matteo, Shawcross rose at the far post to beat Dean Kiely with a powerful header.
Despite the shock of falling behind, Albion continued to press but Koren was frustrated by Simonsen while Miller flashed a shot across the face of goal.
It could have been a costly miss as Stoke then created only their second real opening 10 minutes into the second half when Mamady Sidibe's header from Lawrence's cross was goalbound only to be turned off the line by Koren.
That let-off galvanised Albion and Barnett earned them their reward.
They could have gone on to win the game but Simonsen pulled off a spectacular save from Sherjill MacDonald and James Morrison fired just over the bar.
Miller then wasted a golden chance two minutes from time. He only had Simonsen to beat but blasted his shot straight at the legs of the former Everton keeper.