Wright scored with a diving header from close range after a cross from Macaulay Gillesphey, which earned the Pilgrims a fourth successive win on their travels.
Argyle had fallen behind against the run of play when Killian Phillips opened the scoring for mid-table, injury-hit Shrewsbury early in the second half.
Manager Steven Schumaxcher used all five of his permitted substitutes as they dominated the play and they drew level with a first goal of the season from skipper Joe Edwards in the 68th minute.
It looked like a determined Shrewsbury side would frustrate the Pilgrims and hold out for a draw until Wright came up with his headed goal in the fifth minute of added time.
Argyle lead the table by a point from Ipswich Town, who won 2-1 at home to Port Vale, and Sheffield Wednesday are one more point behind after their 2-1 victory at Bristol Rovers.
Fourth-placed Barnsley, though, lost ground on the top three as they could only draw 0-0 away to Lincoln City.
Macaulay Gillesphey came into the Argyle starting line-up in the only change from the 1-0 Devon derby victory over Exeter, with Saxon Earley dropping down to the substitutes’ bench.
Gillesphey was part of the Pilgrims’ defensive trio alongside James Wilson and Dan Scarr, with Brendan Galloway moving to take up the left wing-back role.
Former Shrewsbury loan signing Sam Cosgrove and Niall Ennis continued to lead the attack for Argyle in their 3-4-1-2 formation, with Finn Azaz in the ‘number 10’ position behind them. Cosgrove spent the first half of last season on loan to the Shrews but scored only two goals in a total of 11 starts and 11 substitute appearances.
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