The home-side grabbed one back through Geovanni a few minutes later but the Reds sealed their 12th away win of the season with a late strike from Kuyt.
Geovanni tried his luck for the Hull midway through the first half, but the Brazilian's rasping half-volley from just outside the box floated narrowly wide of the post.
The visitors opened the scoring on the stroke of half-time. Referee Martin Atkinson awarded a dubious free-kick after George Boetang was adjudged to of brought down Javier Mascherano ten yards outside the Hull box.
Alonso fired in the set-piece which saw the ball deflect off the wall and land back at the feet of the Spain international, who hit a superb first-time effort into the bottom corner of the net to send his Liverpool side into the lead.
Hull were down to ten men on 58 minutes as Folan saw red for kicking out at Martin Skrtel as the pair chased the ball towards the Liverpool goal.
Rafa Benitez's men then doubled their lead as the Hull defence failed to clear a Yossi Benayoun cross, allowing Skrtel to flick the ball into Kuyt, who made no mistake from close range with a neat header through the legs of keeper Boaz Myhill and into the back of the net.
The home side grabbed one back on 73 minutes as substitute Daniel Cousin raced down the left to slip a ball across the box, into the path of an unmarked Geovanni, who gratefully stuck the ball into the back of the net from eight yards.
Liverpool came close to making it 3-1 moments later, as Kuyt played in a curling cross onto the head of Fernando Torres, but the Reds frontman saw his effort from close range come back off the Hull crossbar.
However, the visitors did seal the victory on 89 minutes as a deflected shot from Alvaro Arbeloa was palmed out by Myhill, but only into the path of Kuyt who sent the ball into the back of the net from eight yards to make it 3-1.