Liverpool agreed to sign Gakpo from PSV Eindhoven on 28 December, but the paperwork formalising the £40 million deal took longer to complete.
The Reds said the 23-year-old is now in contention to feature for the first when Jurgen Klopp's side host Wolves in the FA Cup third round.
"Cody Gakpo is officially a Liverpool player after the formalities of his transfer from PSV Eindhoven were successfully concluded on Tuesday," a statement on the club's website said.
Cody Gakpo is officially a Red after the formalities of his transfer from PSV Eindhoven were successfully concluded on Tuesday ?— Liverpool FC (@LFC) January 3, 2023
"The 23-year-old is now formally registered with the Reds and able to begin work with his new club.
"The Netherlands international could now come into contention for Saturday's FA Cup tie with Wolverhampton Wanderers at Anfield."
Gakpo's arrival is well-timed for Klopp after sixth placed Liverpool crashed to a shock 3-1 defeat at Brentford in the Premier League on Monday.
With Liverpool's Uruguay striker Darwin Nunez struggling to live up to the hype that surrounded his £64 million move from Benfica last year, Gakpo is likely to get an immediate chance to shine for his new club.