Hull's 5-3 victory over third-tier Sheffield United in the second FA Cup semifinal, which set up a meeting with Arsenal in the final, was overshadowed by the latest installment in the fight for the league title that still has some way to run.
Seeking a first championship in 24 years, Liverpool is the favourite after taking its winning streak to 10 games thanks to Philippe Coutinho's match-clinching strike in the 78th minute against City.
Tears were shed by Liverpool captain Steven Gerrard as he embraced his teammates and made a rousing speech at the end of an emotional contest that was played to a backdrop of tributes marking the 25th anniversary of the Hillsborough Stadium disaster, in which 96 Liverpool fans died.
"We live with Hillsborough every day — the families and the victims are in our thoughts every day," said Liverpool manager Brendan Rodgers. "It was an inspiration for us, rather than something that held us back."
While third-placed City, which is seven points behind Liverpool with two games in hand, is relying on a slip-up from the rampant leaders, second-placed Chelsea is in charge of its own fate after Demba Ba's goal secured a 1-0 win in south Wales.
Liverpool is two points clear of Chelsea with four games each left but the pair meet at Anfield in two weeks in what is likely to be a defining match in the race for the title. If Chelsea takes anything from that game, City is back in the frame and will capture the league by winning all of its remaining games.
"It is not the end of the Premier League," City manager Manuel Pellegrini said. "The most important thing today is now it is not dependent what we do in the future."
Ba's winner in the 68th minute — a left-foot strike from the edge of the area — ended Swansea's stubborn resistance, with the hosts forced to play most of the match with 10 men following the 16th-minute dismissal of Chico Flores for two yellow cards.
It capped a successful week for Ba, who scored a late goal in Tuesday's Champions League quarterfinal against Paris Saint-Germain to put Chelsea into the last four.
At Wembley Stadium, Hull twice came from behind to reach the FA Cup final for the first time in its 110-year history and stay on course to win its first major trophy. Arsenal beat second-tier Wigan in a penalty shootout on Saturday and is one more win away from ending a nine-year trophy drought.
"All the pressure is on Arsenal," Hull manager Steve Bruce said. "They haven't won anything for years."
Hull was transformed after halftime, with Matty Fryatt equalizing for 2-2 before Tom Huddlestone and Stephen Quinn gave the northeast team what appeared to be a comfortable cushion. Even though Jamie Murphy pulled a late goal back for United, David Meyler quickly responded to secure Hull's first trip to a major final.
Even if Hull beats Arsenal, the story of the season is likely to remain Liverpool's resurgence to the top of the English game, which the club dominated in the 1970s and '80s.
Thousands of Liverpool fans lined the streets outside Anfield hours before kickoff to cheer the team bus into the ground and the sea of flags, scarves and banners in the iconic Kop stand harked back to the club's glory days.
And Rodgers' team keeps on winning, with the platform for the latest victory set up with goals by Raheem Sterling and Martin Skrtel in the opening 26 minutes.
City was blown away at the start but produced a second-half fightback, with David Silva scoring in the 57th before Glen Johnson's own goal five minutes later looked like giving the visitors a point.
Coutinho, though, pounced on City captain Vincent Kompany's poor clearance to curl home a low strike and send the home crowd wild. Jordan Henderson's injury-time red card for Liverpool for a studs-up challenge on Samir Nasri came too late to affect the outcome.
"That is probably the biggest statement we have made so far," Gerrard said. "That was the longest 90 minutes I have probably played in."