Liverpool ease through but Juve end Shamrock's hopes
David Ngog, who scored both goals in the first leg last week, wrapped matters up when he headed Liverpool ahead after 22 minutes following a perfect pass from Cole.
The five-time European champions doubled their lead after 40 minutes when Steven Gerrard cooly wrong-footed the Macedonian side's goalkeeper Martin Bogiatinov from the spot after Fernando Lopes hauled down Ngog.
England midfielder Cole, who moved to Anfield from Chelsea last month, almost scored seconds after half-time when his far post cross hit the bar.
The visitors were totally outplayed and only escaped a much heavier defeat because of some wayward Liverpool shooting, the only minor niggle for new manager Hodgson in his first match in charge at Anfield. "It's nice to be in the next round although, quite honestly, I thought we would be after that good performance in Skopje," said Hodgson afterwards.
"It was important tonight to be disciplined and have a quality performance and we certainly did that."
Shamrock Rovers went out of the competition with their heads held high following a 1-0 defeat away to Juventus.
Rovers' hopes of causing a huge upset in the third qualifying round were all but over after a double from Amauri gave Juventus a 2-0 victory in the first leg at Tallaght Stadium in Dublin last week.
However, they pushed Juventus all the way and only Alessandro Del Piero's 74th-minute strike separated the teams, with the Old Lady winning 3-0 on aggregate.