Oliver Norwood fired home a stoppage-time penalty to sink Wolves and give Sheffield United their first Premier League win of the season.
Fabio Silva fouled George Baldock in the box and after a video assistant referee (VAR) check, Norwood converted from the spot in the 10th added minute.
Archer was clean through on goal and left Wolves defenders in his wake before his shot went in off the crossbar, leaving goalkeeper Jose Sa helpless.
Jean-Ricner Bellegarde scored an 89th-minute equaliser, curling into the top corner, before Norwood restored United’s lead.
The home side put an end to a six-match losing run but remain bottom of the table on goal difference, having taken just one point from 10 games before this long-awaited victory.
United had been the better side in the second half and were rewarded in the dying seconds.
Wolves could have gone ahead in the first half but Matheus Cunha’s effort was tipped over the bar by Wes Foderingham.
Hwang Hee-chan then set himself smartly in the box after Sasa Kalajdzic’s pass but fluffed his shot.
The resulting ball came at Rayan Ait-Nouri too quickly for him to be able to sort his feet out and convert, on a day when Wolves’ five-game unbeaten run in the league came to an end.