Pep Guardiola's side, 2-1 winners at second-place Manchester United on Sunday, won comfortably at the Liberty Stadium to eclipse the mark of 14 successive wins set by Arsenal in 2002.
United bounced back from their derby defeat by beating Bournemouth 1-0, while slip-ups for Liverpool and Arsenal allowed Tottenham Hotspur to steal into fourth place by beating Brighton and Hove Albion 2-0.
"Winning 15 games in a row gives us a lot of confidence. If people are happy watching us, that is the best gift," Guardiola said.
"You suffer and enjoy, like a manager you see many things. When the team play good, we are happy."
City remain 11 points clear of United and have now scored 52 goals in their 17 league games to date as they stride towards what seems like an inevitable title coronation.
Guardiola made four changes to the team that had prevailed at Old Trafford in a game marred by an angry post-match brawl that is being investigated by the Football Association, with Sergio Aguero earning a recall.
Silva broke the deadlock at bottom club Swansea in the 27th minute by flicking in Bernardo Silva's cross from the right.
Kevin De Bruyne's free-kick made it 2-0 shortly after and David Silva added a third early in the second half, neatly chipping Lukasz Fabianski following a give-and-go with Raheem Sterling.
Aguero put the cherry on the cake in the 85th minute by surging away from a posse of defenders and finishing with a low shot.
Romelu Lukaku scored the game's only goal as United got back to winning ways against Bournemouth.
The Belgian striker, who is alleged to have been heavily involved in the derby dust-up, headed in Juan Mata's 25th-minute cross and pointedly failed to celebrate.
"I am pleased with the three points. It was a difficult match," United manager Jose Mourinho told the BBC.
"They had one more day's rest than us, they were fresher than us and mentally fresher because a big match takes more from our players."
- Arsenal drift -
Tottenham jumped from seventh place to fourth courtesy of their 2-0 win over Brighton at Wembley, which arrived courtesy of an overhit Serge Aurier cross and Son Heung-min's fourth goal in four games.
Spurs, for whom Erik Lamela made his first start in 14 months, are four points below third-place Chelsea in the last Champions League berth.
"I think it's fantastic for us," said Spurs manager Mauricio Pochettino.
"Three victories in a row -- so important for us. I'm happy with the performance."
Liverpool are fifth, on goal difference, after a 0-0 draw at home to West Bromwich Albion.
Jurgen Klopp recalled Philippe Coutinho and Roberto Firmino following Sunday's 1-1 derby draw with Everton and thought Dominic Solanke had poached a late winner, only for the goal to be ruled out for handball.
West Brom have now gone a club-record 16 games without a win in all competitions, but the point took Alan Pardew's side out of the relegation zone.
Arsenal dropped to seventh, a point below sixth-place Burnley, after a goalless draw at David Moyes's improving West Ham United.
Second-bottom West Ham almost won it at the death, but Javier Hernandez's shot from Mark Noble's pass hit the bar and bounced just short of the line.
"We had so much of the ball, but it was very frustrating because we could even have lost it at the end," said Arsenal manager Arsene Wenger.
Everton remain unbeaten under new manager Sam Allardyce following a 1-0 win at his former club Newcastle United, Wayne Rooney settling the game with his seventh goal in eight league games.
Newcastle had Jonjo Shelvey sent off in stoppage time for two bookable offences.
Claude Puel enjoyed a successful return to his former club Southampton as in-form Leicester City won 4-1 at St Mary's to notch up a fourth successive win.
Japanese forward Shinji Okazaki scored twice for Leicester, with Riyad Mahrez and Andy King also on target, while Maya Yoshida replied for the hosts.