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 this deserved victory, boasting the crucial second leg at the Liberty Stadium and the possibility of a south Wales Wembley 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 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.
Source: PA
Source: PA