Tonda Eckert made four changes to the team that started on Saturday in the disappointing home defeat by Hull City.
Switching to a 4-2-3-1 formation, the Saints looked more balanced from front to back. Elias Jelert started the match at right back, with Welington recalled at left-back.
Jack Stephens earned a start at centre-back alongside Taylor Harwood-Bellis, displacing both Nathan Wood and Joshua Quarshie.
Wantaway striker, Adam Armstrong, dropped to the bench in favour of Ross Stewart, who scored against Hull at the weekend.
Southampton were ahead early when Léo Scienza's right-footed shot was heavily deflected by Oliver Arblaster into his own net after seven minutes.
Half-Time: Southampton 1-0 Sheffield United
Sheffield United started the second half brightly and almost drew level when Heck's shot was blocked by Elias Jelert six yards out.
Sheffield United keeper, Michael Cooper, did well to keep out a header from Léo Scienza just after the hour mark. Substitute, Adam Armstrong latched on to a through ball by Finn Azaz and crossed deep to the far post, where Scienza headed powerfully but failed to get the ball wide of the keeper.
A mix-up between Harwood-Bellis and Jelert almost handed Sheffield United a point, but Peretz saved Patrick Bamford's shot with his left leg.
Full-Time: Southampton 1-0 Sheffield United
Goalkeeper and Defenders
Daniel Peretz (GK) - 7/10 - Not the most convincing of first-half performances. Punched a cross when he could have caught it, and then needlessly put his team in trouble with a lazy pass out to Elias Jelert. All over the place from an inswinging corner in the second half, forcing Harwood-Bellis into a goal-line clearance. Made up for all of the above when he saved from Patrick Bamford in the last minute of the ninety to preserve the clean sheet.
Elias Jelert (RB) - 7/10 - Started strongly in both defensive and attacking transitions. Made a vital defensive clearance in front of his own goal when United looked destined to score.
Taylor Harwood-Bellis (CB) - 7/10 - As solid as ever in the heart of Southampton's defence. Handed a reprieve for a last-minute mistake by Daniel Peretz.
Jack Stephens (CB) - 7/10 - A good block at the start of the first half against O'Hare's shot ensured that Sheffield United did not get back into the game. Brave on the ball when he needed to be.
Welington (LB) - 8/10 - Involved positively in the build-up to the first goal. His tackle and awareness to slip the ball to Scienza justified his inclusion ahead of Ryan Manning. Welington's pace, bite in the tackle, and willingness to get forward were evident throughout. Replaced by Manning with ten minutes to go.
Midfielders
Flynn Downes (CM) - 7/10 - Won just about everything in the first half. His first-half interception nearly resulted in Finn Azaz putting the Saints two up.
Caspar Jander (CM) - 7/10 - Got around the pitch well, looking more threatening going forward than in recent weeks.
Attackers
Léo Scienza (AML) - 7/10 - A constant thorn in Ki-Jana Hoever's side all evening. Opened the scoring with a deflected effort and continued to create openings throughout the first half. Needlessly booked early in the second-half.
Finn Azaz (AMC) - 6/10 - Wasted two good chances to put the Saints ahead in the first half, but was heavily involved in most of their good work going forward. Replaced by Shea Charles with ten minutes to go.
Tom Fellows (AMR) - 7/10 - Had a very effective game playing further forward tonight. Unlucky not to win a penalty in the first-half when he was brought down when through on goal. Always a threat on the counter-attack, but his end product did not match his dribbling.
Ross Stewart (CF) - 7/10 - Held the ball up well and brought others into the game. A threat in the box. Withdrawn after an hour and replaced by Adam Armstrong.
Substitutes:
Adam Armstrong - 6/10
Shea Charles - 6/10
Ryan Manning - 6/10
Jay Robinson - 6/10