At the other end Harrison Burrows saw his shot from distance deflected wide for a corner, from which the Swans had to scramble clear as Tyrese Campbell threatened to sneak in to convert at the near post.
After that swarming Swansea start, the remainder of the half was a more even tussle at it was goalless at the break.
Swansea threatened again early after the break, with Josh Key and Cameron Burgess having efforts blocked after Cooper had parried another excellent delivery from Tymon free-kick.
Eom then sprung the United offside trap but overhit his cross, before Cooper gathered in Tymon’s low delivery as the hosts enjoyed a good spell as the hour mark approached.
Zeidane Inoussa and Widell were summoned from the bench in place of Eom and Yalcouye.
Inoussa was soon involved, heading a Key cross against Cooper from an acute angle at the far post.
But the Swans got their reward with 24 minutes to play when Ronald drove play forward and found Inoussa. The Sweden forward sent a low cross in that Cullen could not connect cleanly with, but the Brazilian smashed the loose ball into the net from just inside the area.
Widell nearly doubled their lead eight minutes later, but slid the ball agonisingly wide after Cullen and Inoussa had found his run into the area.
Depsite missing that chance the hosts saw out the remainder of the game in relative comfort as they put a first three points of the season on the board.
Swansea City: Lawrence Vigouroux; Josh Key, Ben Cabango (captain), Cameron Burgess, Josh Tymon; Malick Yalcouye (Melker Widell 62), Ethan Galbraith, Gonçalo Franco; Ronald (Jay Fulton 90), Jisung Eom (Zeidane Inoussa 62); Liam Cullen (Bobby Wales 90).
Unused Substitutes: Andy Fisher, Žan Vipotnik, Kaelan Casey, Ollie Cooper, Sam Parker.
Sheffield United: Michael Cooper, Tyler Bindon, Gustavo Hamer, Callum O’Hare, Harrison Burrows, Djibril Soumare (Andre Brooks 76), Jack Robinson (captain), Tyrese Campbell (Tom Cannon 63), Louie Barry (Ryan One 76), Femi Seriki, Sydie Peck.
Unused Substitutes: Adam Davies, Sam McCallum, Ehije Ukaki, Sam Curtis, Rhys Norrington-Davies, Owen Hampson.
Referee: Thomas Kirk
Attendance: 15,816