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.
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.
Silva broke the deadlock at bottom club Swansea in the 27th minute by flicking in Bernardo Silva's cross from the right.
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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.
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.
Anthony Martial fired over an open goal in the second half after Asmir Begovic foiled Lukaku and United substitute Marcus Rashford rattled the bar, but Lukaku's goal was enough for Jose Mourinho's side.
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.
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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.
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 from 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.
Arsene Wenger's Arsenal, who hit the post through Alex Iwobi in the first half, have now gone three league games without victory.
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.
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.
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.