The winner, worthy of being in the conversation for Championship goal of the season, extended Stoke’s unbeaten run to five matches and added to Sheffield United’s woes, as they slumped to a third successive defeat in all competitions.
There was a minute’s pre-match applause and Stoke wore black armbands in tribute to former Potters forward Harry Burrows, who died earlier in September at the age of 85.
Both sides gave their supporters reasons to keep the ovations going in a high-energy start to the game at bet365 Stadium, with Liam Kitching heading narrowly wide for the Blades before Djibril Soumare tested visiting goalkeeper Michael Cooper with a shot from outside the box.
A heavy touch from Patrick Bamford, as he attempted to bring a cross down inside the box, meant Sheffield United’s clearest opening of the first half was allowed to dribble into the grasp of goalkeeper.
Stoke’s Lamine Cisse went on to drag an effort wide and Ben Johnson thumped a shot into the arms of keeper Cooper in first half that lacked any of the finishing quality that would transform the game in the second half.
Former Leeds United midfielder Phillips produced the smartest finish of the afternoon when he put the visitors ahead with his first goal for the Blades since joining them for a second loan stint from Manchester City in the summer.
He looked as if he would lift his free-kick from wide on the left-hand side into the box in search of a team-mate, but he instead bent his set-piece around the feeble one-man defensive wall and into the bottom corner instead.
Stoke pressed for a leveller and threatened a number of times before McGlynn eventually made it 1-1, letting loose from 30 yards after being allowed time and space to line up his fierce shot.
While Stoke looked most likely to go on and win it, Sydie Peck wastefully put a chance wide at the other end before Gallagher earned the Potters all three points with his memorable goal.