Fulham boss Jol insisted he was not feeling the heat after last weekend's loss to Manchester United, but another heavy defeat is sure to raise more questions over his future.
Liverpool were scintillating going forward and Steven Gerrard's dangerous free-kick went in off Fernando Amorebieta to put them in front, before Martin Skrtel headed home a Gerrard corner to score his first goal since he struck in a 4-0 victory against the same opposition last December.
Suarez made it three before half-time and then helped himself to his eighth Premier League goal of the season with another clinical finish shortly after the break as Fulham caved in.
Brendan Rodgers' side could have won by a more emphatic margin, but they have now won three in a row at Anfield - scoring 11 goals in the process - and Chelsea's home draw West Brom ensured they moved up place to second, just two points adrift of leaders Arsenal.
Fulham, on the other hand, have lost their last four in all competitions and are languishing perilously close to the relegation zone.
There were three changes to the Liverpool side that lost at Arsenal last weekend, with Philippe Coutinho and Daniel Agger making their first starts since September. Glen Johnson also returned as Jon Flanagan, Kolo Toure and Mamadou Sakho missed out.
Defender Elsad Zverotic was handed his first Premier League appearance for Fulham and ex-Liverpool midfielder Alex Kacaniklic also started in place of the suspended Sascha Riether and Bryan Ruiz, who dropped to the bench.
Liverpool started brightly and they were in front after 22 minutes when Gerrard whipped in a free-kick from the right that struck Amorebieta and bounced inside the near post.
There was more misery for Amorebieta three minutes later and Gerrard was once again the architect as Skrtel shrugged off the Venezuela international before planting a bullet header beyond the helpless Maarten Stekelenburg.
Fulham were on the ropes and Stekelenburg produced a magnificent save to deny the influential Coutinho, but the Dutch goalkeeper was beaten again nine minutes before the break when Suarez raced onto Jordan Henderson's fine through ball before finishing unerringly with his right foot.
Liverpool carried on where they left off at the start of the second half and Stekelenburg produced a fine saves to deny Suarez and Daniel Sturridge.
But Suarez added a fourth nine minutes after the interval when Gerrard robbed Kieran Richardson to play in the Uruguayan, who fired inside the near post.
The over-worked Stekelenburg then denied Coutinho again with Fulham at sixes and sevens at the back and, despite creating several more clear-cut opportunities, Liverpool were forced to settle for a four-goal winning margin.
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