The arrivals were due to include 27 children and 79 women, Malaysian media said, the remaining 1,370 adult men.
Rohingya make up the majority of Myanmar refugees and asylum seekers in Malaysia, but Kuala Lumpur has not responded to an AFP query about the number onboard.
Anwar announced the repatriations in July, just days after the detention of more than 100 Rohingya asylum seekers in Kuala Lumpur amid rising xenophobia against refugees, particularly the Rohingyas in Malaysia.
News of the returns sparked widespread concern, as members of the ethnic group fled Myanmar en masse following a military crackdown starting in 2017, which rights experts say amounted to genocide.
Malaysia insists all returnees agreed to voluntary repatriation from detention centres, where Myanmar says they had completed sentences for immigration offences.
But monitor organisations such as Fortify Rights report dire conditions in the facilities.
The UN refugee agency said this past week its staff were not allowed to meet returnees to confirm whether migrants’ decisions “were genuinely free and informed”.
Myanmar state television has broadcast footage of the migrants on the journey home, being led in aerobics classes on a ship deck and wearing T-shirts saying: “Happy voyage back to motherland”.