Basham, who missed a golden first-half chance, made amends when he drilled home after fine work by ex-Leeds forward Billy Sharp.
An eventful game saw home keeper Kiko Casilla red-carded for committing a professional foul and bringing down Sharp outside of his area, with Pontus Jansson moving into the goal with the hosts having used all three substitutes.
But the decisive moment came from Basham as the Blades avenged their defeat at Bramall Lane earlier this season.
The first period was a half which saw Leeds exhibit elements of the verve and poise which has seduced many football lovers this season, but fail to forge a breakthrough, which would have been deserved on the balance of play.
By contrast - and by their own high standards - the Blades produced an uncharacteristically sloppy performance going forward in the main, although a spell of pressure in the build-up did at least give them a semblance of hope.
It produced arguably the best chance too with a defence-splitting pass from David McGoldrick sending Basham clear in the last action of the half, but the utility man, lining up in midfield, waited for support instead of unleashing an effort when offered a clear sight of goal after rounding Kiko Casilla and his pull-back was cleared.
Much of the action was at the other end, with Pablo Hernandez again orchestrating play with aplomb and outshining the likes of John Fleck and Oliver Norwood.
A stirring home atmopshere, with a minutes' applause taking place beforehand to celebrate the life of the late Bill Fotherby, was the prelude to a rousing opening from the hosts.
A terrific block from John Egan denied Patrick Bamford after Luke Ayling and Hernandez opened up the Blades down their left before Jack Harrison spurned a superb opportunity, with his close-range volley flying over after more classy work from Hernandez and Mateusz Klich.
The buttons were being pressed by Leeds, with the Blades then suffering a let-off after a poorly-timed challenge from Geroge Baldock - after he lost control of the ball - on Pontus Jansson yielded a caution instead of a red card, with the tackle being similar to the one which saw Rotherham's Will Vaulks dismissed in their fixture at Bramall Lane seven days earlier.
The next to threaten for Leeds was Patrick Bamford, whose fierce shot from a different angle flew over before Tyler Roberts spurned a presentable chance, heading off target with left unmarked following Gjanni Alioski's left-wing centre.
Another polished Leeds move then ended with Alioski firing over, with the Blades - after a subdued half - afforded a spot of hope with a mini-spell of pressure before the break.
A Fleck corner cuased consternation with Egan's shot eventually blocked before an alert challenge from Jansson thwarted Sharp ahead of Basham spurning his big opportunity.
Relentless rain on the resumption made the pitch decidedly sticky and chances were harder to come by as flowing football became difficult.
The Blades, who started the second half more convincingly than they began the first period and had a half-chance when Ollie Norwood drove over with Kalvin Phillips having earlier fired an effort into the Kop.
Leeds then went mighty close to yielding a breakthrough when Roberts' angled shot hit the outside of the post after a clever pull-back from byline by Klich.
Sharp was then cautioned for catching Jansson as a breakthrough still stubbornly refused to arrive until Basham's moment of atonement.
O'Connell's punt forward was flicked on by McGoldrick with Sharp hoodwinkinng his good friend Copper to cleverly slip in the supporting Basham, who surged forward and drilled an inch-perfect low shot beyond the reach of Casilla.
Leeds threw on Jack Clarke and Stuart Dallas in a bid to wrestle back the initiative and the hosts went desperately close when Jansson fired wide from Clarke's pull-back.
Jansson's eventful game continued when he donned the keeper's gloves after Casilla saw red - but the points were reserved for the Blades.
Leeds United: Casilla, Ayling (Dallas 77), Jansson, Cooper; Hernandez, Phillips, Klich (Clarke 77), Alioski; Harrison (Douglas 57), Roberts, Bamford. Substitutes unused: Peacock-Farrell, Berardi, Shackleton, Gotts.
Sheffield United: Henderson; Cranie, Egan, O'Connell; Baldock, Basham (Lundstram 90), Norwood, Fleck, Stevens; McGoldrick (Dowell 88), Sharp. Substitutes unused: Hogan, Coutts, Stearman, Duffy, Moore.
Referee: David Coote (Notts).