Both sides had squandered numerous chances in a scrappy Europa League round of 32 encounter but Liverpool's superior quality was eventually rewarded by the hard-working Kuyt.
David Ngog, Joe Cole and Raul Meireles had also gone close in front of a sell-out 42,949 crowd which further underlined the gathering momentum of Dalglish's new regime.
Sparta, backed by 3,000 of their own raucous fans, contributed fully to an eventful, if low-quality, clash but were eventually undone as Kuyt profited from a late corner.
Liverpool captain Steven Gerrard, absent for the two previous games with a groin injury, was not risked and his dynamism and precision was missed throughout.
Christian Poulsen, whose partner had a baby earlier in the week, made little impact in a central role while Cole, starting for the first time under Dalglish, was quiet on the left.
The former Chelsea midfielder looked determined to kickstart his faltering Reds career, however, and broke clear early on to send over a looping cross which clipped the bar.
The opening stages were lively and suggested the game, which ultimately lacked quality, would be far more eventful than the drab first leg.
Liverpool appealed for a penalty in the second minute when a Meireles free-kick bounced and caught Manuel Pamic on the hand but nothing was given.
At the other end Libor Sionko headed over from a Kamil Vacek cross and a dipping shot from Vaclav Kadlec narrowly missed the target.
Both goalkeepers were tested by headers, Jose Reina first claiming a firm effort from Sinoko.
Sotirios Kyrgiakos then went close to handing Liverpool the lead when he rose to meet a Meireles corner but Jaromir Blazek saved well to his right.
Reina had a scare when he failed to claim a corner but Sionko was unable to take advantage and Meireles missed a good chance when he scooped a cross from the rampaging Martin Kelly over.
Blazek had an uncomfortable moment when he spilled a long-range shot from Cole, who had cut in from the left, but Liverpool could not capitalise.
Ngog tried his luck from distance as the ball dropped invitingly but miskicked as he turned to shoot.
Ngog had a better chance moments later as Kuyt cut inside to tee him up but the Frenchman chipped tamely at Blazek, who almost gifted a rebound opportunity to Cole but recovered.
Lucas also saw a 25-yard shot drift narrowly wide just before the interval.
Liverpool were forced to withdraw one of the first half's better players in Kelly at the start of the second period, but were able to replace the right-back with the experienced Jamie Carragher.
The hosts started brightly after the restart with Meireles volleying narrowly wide and then creating another attack with a superb long ball for Kuyt.
Kuyt raced on into the area and pulled back for Ngog but the lone striker again fluffed his shot.
Sparta also continued to threaten but Reina got down well to cut out a cross from Ondrej Kusnir.
Liverpool met more frustration as they tried to prise an opening with Kuyt having a shot blocked and Cole heading wide midway through the second half.
Liverpool pressed further with Meireles playing in Ngog but Blazek denied him.
Kyrgiakos was forced off for treatment after a clash of heads with Tomas Repka left him with a nasty cut above the eye.
He returned just as defensive partner Daniel Agger was forced off with injury and replaced by Martin Skrtel.
Kuyt, along with Meireles, had been one of Liverpool's most forthright players in the second half and his efforts were rewarded with his winner.
The Dutchman jumped above the defence five minutes from time to head firmly past Blazek and put the Reds into the last 16.
Cole should have added a second when he raced through on goal in injury time but sliced his shot wide.
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