Francesco Guidolin's future as Swansea manager looks bleak after Liverpool fought back to win 2-1 at the Liberty Stadium.
Guidolin admitted before the game that his job was in jeopardy if Swansea, winless since the opening day of the season and just above the relegation zone, did not beat a Liverpool side who are now second in the Premier League after a fourth straight victory.
The Italian saw his side produce their best performance of the season, but Leroy Fer's early tap-in was eventually overturned by a Roberto Firmino header and a James Milner penalty six minutes from time.
Liverpool took time to get into their elegant stride, but they dominated the second period and another three points will ensure that the Reds mark Jurgen Klopp's first anniversary at Anfield next weekend in a strong league position.
Further doubt on Guidolin's tenuous position appeared to be cast before kick-off when chairman Huw Jenkins failed to mention the Italian by name in his programme notes.
Swansea have been linked with Ryan Giggs and former United States manager Bob Bradley to become their next manager, but the hosts began brightly.
Record signing Borja Baston, making his first Premier League start with Fernando Llorente sidelined by a rib injury, headed over from 10 yards when left unmarked.
Klopp had fielded questions on Liverpool's vulnerability at set-pieces when appearing in the Sky television studio earlier in the week.
But it seemed his defenders had not been listening to him as Angel Rangel rose unopposed at the far post to meet Gylfi Sigurdsson's corner.
The header was touched on by some combination of Mike van der Hoorn and Liverpool's Dejan Lovren for Fer to tap home his fourth goal of the season from a yard out and ensure Liverpool's wait for a clean sheet in the league this term continues.
Liverpool were struggling to find their usual rhythm, pressed into mistakes while playing from the back and unable to launch any attacks of substance.
Fer drove over from 25 yards as Liverpool defenders dawdled on the ball once more and the Reds' frustration grew as Adam Lallana limped off to be replaced by Daniel Sturridge.
Sigurdson's eye for a pass was causing Liverpool huge problems with his ball over the top of finding a stretching Jack Cork, who could not get enough power in the shot to test Loris Karius.
Borja should then have opened his Swansea account when Sigurdsson's free-kick set him up from eight yards, but the Spaniard planted his header wide.
Liverpool finally roused themselves in the closing stages of the first half after Sturridge was booked for simulation in the penalty area.
But Sadio Mane and Roberto Firmino were foiled by brave blocks from Jordi Amat and Fer, while Karius held Sigurdsson's free-kick at the second attempt at the other end.
Liverpool stayed on the front foot after the restart, with Firmino forcing a first save from Lukasz Fabianski and Amat clearing as Mane's shot squirmed across an unguarded six-yard box.
Their pressure was rewarded after 54 minutes when Jordan Henderson lifted a ball into the box after Philippe Coutinho's free-kick had been blocked and Firmino nodded home his fourth goal of the season.
Coutinho curled home as Liverpool sought to make their superiority count and Nathaniel Clyne fired straight at Fabianski.
But the telling moment came when substitute Modou Barrow skied a clearance and Rangel barged over Firmino as the ball dropped, allowing Milner to dispatch his third penalty in two games.
Van der Hoorn's late miss meant there was no way back for Swansea.