Late Kuyt goal sinks Sparta
Ajax Amsterdam and Benfica, two other big names in European football, also went through, the Dutch beating Anderlecht 2-0 and the Portuguese winning by the same score at VfB Stuttgart.
PSV Eindhoven beat Lille 3-1 to progress from a meeting of Dutch and French league leaders while Russian pair Zenit St Petersburg and Spartak Moscow overcame Swiss opposition in frigid conditions.
Germany captain Michael Ballack was among the scorers as Bayer Leverkusen beat Metalist Kharkiv 2-0 to complete a 6-0 aggregate victory over the Ukrainians.
Big-spending Manchester City qualified with a 3-0 win over Aris Salonika, helped by two goals in the first 12 minutes from recent Bosnian signing Edin Dzeko after a goalless first leg. Yaya Toure added the third with a deflected shot.
They will face Dynamo Kiev, who beat Besiktas 4-0 with Andriy Shevchenko among the scorers for an emphatic 8-1 aggregate scoreline.
Italy's interest ended when Villarreal beat Napoli 2-1 following last week's goalless draw. The Spanish side will now meet Leverkusen.
In Kenny Dalglish's first European home match as Liverpool manager - the club were banned from Europe during his previous stint - Kuyt headed in from a corner after 86 minutes to secure victory over the Czechs with extra-time 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.
Liverpool will face Braga, who beat Lech Poznan 2-0 for a 2-1 aggregate win, in the next round.
Serb Miralem Sulejmani scored two early goals as four-times European champions Ajax completed a 5-0 win over their Belgian opponents and former UEFA Cup winners Anderlecht.
Benfica's Eduardo Salvio and Oscar Cardozo, two members of their South American contingent, completed a 4-1 aggregate win for former European champions Benfica over Stuttgart.
The Portuguese will next play Paris Saint-Germain after the French side went through on away goals against BATE Borisov.
Pierre-Alain Frau gave Lille a half-time lead at PSV but Balasz Dzsudzsak, Jeremain Lens and Marcelo replied for the Dutch in an 18-mi