City's victory sets up a play-off semi-final showdown with Nottingham Forest.
Swans went into the game knowing they had no chance of automatic promotion after champions QPR received no points deduction from the Football Association, a decision which it is understood Swans chairman Huw Jenkins will take no further action against despite the possibility of appeal.
What is for sure after months of uncertainty, is that City will now head into the play-offs in confident mood after today's deserved victory, boasting the crucial second leg at the Liberty Stadium and the possibility of a south Wales showdown with Cardiff, who now face Reading.
Stephen Dobbie opened the scoring on the half hour with a free-kick whipped in at pace from an acute angle which evaded everyone.
Blades defender Shane Lowry then saw red for handling Scott Sinclair's goalbound effort on the line, with the England Under-21 winger stepping up to score his 24th goal of the season from the spot.
Dobbie then grabbed his second of the afternoon and Leon Britton rounded matters off late on against his former club, to leave a youthful and relegated United second from bottom in the final standings on goal difference.
City boss Brendan Rodgers made two changes to his XI, restoring star names Sinclair and Fabio Borini after they were rested for last weekend's impressive 2-0 win at Millwall.
Blades counterpart Micky Adams - who spent 13 days as Swans boss in 1997 - made three changes to his line-up, most notably handing a debut to goalkeeper George Long.
And the 17-year-old, a star of the club's successful FA Youth Cup side, was called upon after just 70 seconds to turn Dobbie's snap-shot around a post.
The hosts looked most likely to break the deadlock and, at the end of a well-worked move, Sinclair saw two shots blocked before Nathan Dyer then fired over the crossbar.
Long again produced another great save from Dyer before Lee Williamson, played in by Stephen Quinn, wasted a clear-cut chance to hand United the lead, seeing his shot turned out by Dorus de Vries.
That could have changed the complexion of the clash but, instead, it was left for Dobbie to strike first after 30 minutes with a set-piece wide from the left.
Sinclair and Dobbie both missed chances to extend the lead before the break, but it did not take long after the restart to do so.
Sinclair's shot was deliberately blocked by the hand of Lowry, who saw red for his actions, and the former Chelsea winger stepped up to stroke home the resulting penalty.
It was 3-0 on the hour as Dobbie rounded off a sweeping move which involved Swans centre-back Ashley Williams.
Williams was in the mood and his stunning effort from distance came agonisingly close.
The final 20 minutes were played out amid an end-of-season tempo, with United boss Adams handing valuable game time to youngsters Terry Kennedy, Daniel Philliskirk and Matty Harriott as the club prepare for life in League One alongside city rivals Sheffield Wednesday.
But there was still time for Britton to add a well-taken fourth in the 90th minute.