In Kenny Dalglish's first European home match in charge of Liverpool, Kuyt headed in from a corner after 86 minutes to secure victory over Sparta Prague.
Dirk Kuyt headed a late goal to send Liverpool into the last 16 of the Europa League with a 1-0 win over a defiant Sparta Prague at Anfield on Thursday.
PSV Eindhoven beat Lille 3-1 to progress from a meeting of the Dutch and French league leaders and Russian pair Zenit St Petersburg and Spartak Moscow overcame Swiss opposition in frigid conditions.
Michael Ballack was among the scorers as Bayer Leverkusen beat Metalist Kharkiv 2-0 to complete a 6-0 aggregate win over the Ukrainians.
In Kenny Dalglish's first European home match in charge of Liverpool, Kuyt headed in from a corner after 86 minutes to secure victory over the Czechs just as extra time was looming.
"We deserved it, we had a lot of chances," Dalglish told Britain's Channel 5 television. "Even after we scored we had a couple. Tough opposition but they hung in, great credit to the lads for their determination and effort to get a result."
The first leg had ended goalless in Prague.
Pierre-Alain Frau gave Lille a halftime lead at PSV but Balasz Dzsudzsak, Jeremain Lens and Marcelo replied for the Dutch in an 18-minute spell as they won 5-3 on aggregate.
Russian champions and 2008 UEFA Cup winners Zenit St Petersburg came back from the brink of elimination to beat Young Boys Berne 3-1.
Leading 2-1 from the first leg, the Swiss went ahead when defender Ammar Jemal headed in after Zenit failed to clear a corner.
In a game which kicked off in a temperature of minus 9 Celsius -- slightly warmer than predicted earlier this week -- Danko Lazovic and Sergei Semak replied either side of halftime to level the aggregate scores.
Roman Shirokov added the third for Zenit who survived a scare when Henri Bienvenue hit the post.
Spartak Moscow drew 1-1 at home to FC Basel in even colder conditions -- around minus 15 -- to go through 4-3 on aggregate.
Edu scored from close range in stoppage time to give Rangers a 2-2 draw at Sporting and send the Scottish side through on away goals to the last 16 where they will face PSV.
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