Reds held by Stoke as Spurs win again
Rodgers has yet to oversee a Premier League home win in four attempts since taking charge of Liverpool in the close season, and the Reds' barren run in front of their own supporters has now stretched to six matches in all competitions.
The hosts hit the woodwork three times, but Liverpool were unable to break down Stoke's stubborn defence and Rodgers' side remain in 14th place after just one win in their first seven league matches.
Liverpool, whose only home success under Rodgers came in the UEFA Europa League against FC Gomel, are already eight points off the top four and, more worryingly for their manager, they lie only three points above the relegation zone.
After a slow start under new manager Andre Villas-Boas, Spurs have climbed to fifth place and they were always in control against struggling Villa, who slip to 16th position.
Villas-Boas, who ended USA goalkeeper Brad Friedel's run of 310 consecutive Premier League starts by selecting recent signing Hugo Lloris instead, saw defender Caulker break the deadlock in the 58th minute when he deflected in Jermain Defoe's shot.
England winger Lennon sealed the points with a surging run and cool finish in the 67th minute.
Point saved for Saints
Elsewhere, Southampton defender Jose Fonte headed a late equaliser to salvage a 2-2 draw against Fulham in a dramatic finish at St Mary's.
Nigel Adkins' side looked set for their sixth defeat in seven league matches this season when Kieran Richardson put Fulham 2-1 ahead with just two minutes to go.
But Portuguese centre-back Fonte, who had earlier given Saints the lead before a Jos Hooiveld own goal drew Fulham level, came to the rescue as he popped up at the far post to nod home a free-kick in added time.
Fonte's goal lifted Southampton out of the relegation zone at Reading's expense, while Fulham remain in ninth place.