A message posted on Twitter at 3.30 a.m. (local time) suggested that the international group of hackers who call themselves Anonymous carried out the cyber-attack, public broadcaster NHK reported.
They apparently used a tactic called a distributed denial-of-service attack (DDoS) that involves overwhelming a website by sending massive amounts of data.
Abe's office and the Tokyo Metropolitan Police Department have an investigation.
A number of Japanese websites have recently come under suspected attacks by Anonymous.
The health, labour, and welfare Ministry's website was inaccessible for three days in November.