Bellamy flourished on his second full debut for the club, having rejoined the Reds from Manchester City in the summer.
The 32-year-old broke the deadlock in the seventh minute with a neat finish from an awkward angle and performed with aplomb throughout, including a superb 30-yard free-kick that rattled the crossbar.
Brighton beat Sunderland in the previous round but they never looked likely to add another Premier League scalp in the first half.
However, the Seagulls were greatly improved after the break as they desperately attempted to maintain their unbeaten start to life in their new stadium.
Craig Noone’s superb curling effort hit the crossbar shortly after the restart and Ashley Barnes scored a late penalty, but Kuyt had already put the tie out of reach.
The Holland international was put free by Maxi Rodriguez with less than 10 minutes remaining and finished strongly to the delight of the 2,403 visiting Liverpool fans, who also saw talisman Steven Gerrard make his first appearance since March.
Kenny Dalglish made seven charges to the side that started Sunday’s 4-0 defeat at Tottenham, with summer signings Sebastian Coates and Bellamy both handed their first Liverpool starts this season.
Brighton, who face Leeds in the Championship on Friday evening, also made alterations, with Gus Poyet handing a full debut for Steve Cook as part of his five changes from the side beaten 1-0 at Leicester.
The third-round encounter was the pair’s first meeting in more than 20 years and one that Liverpool started strongly.
Jay Spearing blazed a long-range shot over in the fourth minute and only a last-ditch tackle prevented Rodriguez prodding goalwards after a melee in the box.
Liverpool’s pressure paid dividends in just the seventh minute when Luis Suarez poked the ball through to Bellamy, who finished neatly from a tight angle.
The visitors continued to attack after breaking the deadlock and Kuyt saw a fierce shot blocked by Gordon Greer.
The Dutchman almost snuck in at the back post soon after and then had a shot cleared off the line from the resulting corner.
The visitors’ attacking intent had Brighton pinned back and they came agonisingly close to doubling their advantage in the 26th minute.
Poyet’s defence was being given a torrid time by fellow Uruguayan Suarez, who shot inches wide when up against goalkeeper Casper Ankergren.
The 24-year-old again came close six minutes later, glancing an inswinging Bellamy free-kick just wide of the far post.
The Welshman saw a rasping 30-yard free-kick rattle the crossbar as half-time approached, before Ankergren was forced to tip Spearing’s superb left-footed drive onto the post.
Despite Liverpool’s dominance of the opening period, the Seagulls almost equalised with their first attack of the game but Martin Kelly’s last-ditch tackle in the six-yard box thwarted the home side.
They looked rejuvenated after the break and two minutes in Noone’s 25-yard curling effort hit the crossbar.
Jose Reina was well beaten by the former Liverpool junior but was on his toes to save from Matthew Sparrow from close range shortly afterwards.
Brighton continued to press forward but saw attacking threat Will Buckley forced off in the 60th minute with a knock, resulting in former Spain international Vicente Rodriguez coming on for his debut.
Liverpool made a change of their own with 16 minutes remaining as Gerrard made his long-awaited return from a groin injury in place of Suarez.
The south coast side continued to search for an equaliser but Kuyt seemingly put the match out reach with less than 10 minutes remaining, latching onto Rodriguez’s pass to beat Ankergren.
Substitute Barnes pulled one back from the spot in the 89th minute after Jamie Carragher was adjudged to have fouled Vicente, although the goal proved a mere consolation.