Five-time European champion Liverpool beat Toulouse 4-0 to complete its qualification for the group stage of the Champions League on Tuesday.
Netherlands striker Dirk Kuyt scored twice in the last three minutes for Liverpool, which lost last season's final 2-1 to AC Milan. Earlier goals by Peter Crouch and Sami Hyypia added to a 5-0 aggregate win in the third-round qualifying matches.
The Reds will be joined in Thursday's draw for the lucrative group stage by Lazio and Rangers.
Lazio won 3-1 at Dinamo Bucharest to round off a 4-2 aggregate victory. Rangers beat fierce rival and Scottish champion Celtic into the group stage by drawing 0-0 at Red Star Belgrade to complete a 1-0 win.
Sevilla, which has won the last two UEFA Cups, was set to play AEK Athens but the match was postponed to Monday because of the death of Sevilla midfielder Antonio Puerta.
Puerta died on Tuesday three days after collapsing on the field during his team's Spanish league match against Getafe. Doctors said the 22-year-old Puerta had a heart attack in the locker room, and at least one more in the emergency room of a Seville hospital.
Liverpool already led 1-0 through Andriy Voronin's goal in the first match two weeks ago, and went close to extending its lead through a third-minute shot from a free kick by John Arne Riise that the goalkeeper saved.
Crouch, who has not started a match in Liverpool's unbeaten start to the Premier League season, put his team ahead on the night in the 19th. The England striker flicked in Kuyt's cross to the far post with the outside of his right boot.
"It was important to get through to the next round but it was nice for me to get my first goal of the season," Crouch said.
"It was important we started the season well. If we progress the way we've started, we're looking good."
Hyypia, who was captaining the home side in the absence of the injured Steven Gerrard and Jamie Carragher, stretched the advantage in the 49th with a free header from a corner.
Kuyt then completed the victory with a low shot in the 87th and a flick over the goalkeeper in injury time.
Lazio's place in the competition was in doubt after it drew 1-1 at home two weeks ago, and the Italian side's chances were hit further when it trailed on Florin Bratu's 29th-minute goal.
However, two second-half goals by Tommaso Rocchi and one by Goran Pandev carried Lazio into the group stage, where it will join Italian rivals Inter Milan, AC Milan and AS Roma.
Rangers made it through to the draw after goalkeeper Allan McGregor made two key saves from Igor Burzanovic and one from Ognjen Koroman to preserve the visitor's aggregate lead.
The Serbian champions were unable to break down a defense protecting a 1-0 lead from Nacho Novo's 90th-minute score at home two weeks ago.
Rangers had two best chances in the second half, with Jean-Claude Darcheville's shot flying past the left post and Lee McCulloch's low shot being saved by goalkeeper Ivan Randjelovic.
Half the 32 places in the group stage are already filled by sides including the winners of Europe's top national leagues. Another 11 places will be filled on Wednesday when more third qualifying round matches are completed.
Four-time European champion Ajax is the high profile club most at risk of elimination from the qualifying rounds. The Dutch club trails 1-0 ahead of its match at Slavia Prague.
Two-time runner-up Valencia holds a 3-0 lead going into its home match with Elfsborg, while former winners Celtic and Benfica are also in action.
Sevilla leads AEK 2-0 ahead of their rescheduled match.