Rangers, beaten finalists in 2008, needed an equaliser at the death to prevail at Sporting Lisbon, while Manchester City's expensive squad eased past Greek representatives Aris.
Liverpool enjoyed the lion's share of possession against the Czech side but, as in the first leg, they struggled to make their dominance count.
Raul Meireles blazed over wastefully from an inviting Martin Kelly cross in the first half, while French striker David Ngog was one of several Liverpool players frustrated by Sparta goalkeeper Jaromir Blazek in the second.
With extra-time beckoning, Liverpool won a corner on the left-hand side that Meireles curled into the box and Kuyt darted away from the goal-line to cushion an intelligent header into the bottom-right corner.
Read more of the AFP story at the Herald Sun ....
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