Japan has agreed to ease some sanctions against North Korea once the probe had been reopened and will consider providing humanitarian aid depending on how the investigation progresses, Chief Cabinet Secretary Yoshihide Suga said separately.
"Our job will not end until every parent can embrace their children with their own arms," Mr Abe told reporters. "This is a first step toward an overall resolution."
North Korea promised in 2008 to reopen the probe - but it never followed through. It also reneged on promises made in multilateral talks aimed at ending its nuclear weapons programme and declared the negotiations over.
The agreement on the abductees probe comes at a time of regional concerns that Pyongyang may be preparing for a fourth nuclear test in contravention of UN sanctions.
Asked whether Japan's actions meant Tokyo was out of step with Washington and Seoul, Mr Suga told a news conference: "It's impossible. This agreement covers sanctions that Japan imposed on its own.
"It is not related to UN sanctions."
Suga added that Japan would keep pressing for a "comprehensive resolution" to abduction issues.