Abbas has granted the pardon to Lieutenant Colonel Osama Mansour, 49, following his online post criticising the Palestinian president, according to the Independent Commission for Human Rights.
Abbas's office had not commented.
Mansour was public relations director at the Palestinian military liaison unit, which coordinates security with Israel.
He was arrested 10 days ago and sentenced by a military court to a year in prison for disobeying orders against publicly expressing political opinions.
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Abbas signed the Oslo accords along with late Israeli statesman Peres, who died on September 28, and has called him a "brave" partner for peace.
Peres won the 1994 Nobel Peace Prize jointly with Yitzhak Rabin and Palestinian leader Yasser Arafat for his role in negotiating the Oslo accords, which envisioned an independent Palestinian state.
The hawk-turned-dove also served twice as Israeli prime minister and was president from 2007-2014.
But while Peres is hailed in the West as a peacemaker, many in the Arab world, including among the Palestinians, regard him a "war criminal".
Peres was prime minister in 1996 when more than 100 civilians were killed while sheltering at a UN peacekeepers' base in the Lebanese village of Qana fired upon by Israel.
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