Boko Haram had held the town near the Niger border since November. Ten Chadian soldiers were killed and 20 wounded in the offensive that began on Sunday, the source said.
A Niger official in Diffa, located across the border from Damasak, confirmed the town had been retaken after heavy fighting.
The offensive, which followed a sustained build-up of troops in southern Niger, opened up a new front in regional efforts to wipe out the Islamist group, whose six-year insurgency has spread across borders.
It came after Boko Haram leader Abubakar Shekau pledged allegiance to the Islamic State group in an audio message at the weekend.
More than 13,000 people have been killed and some 1.5 million made homeless in the Boko Haram conflict since 2009 and the regional coalition is a reflection of regional fears about security following cross-border attacks.