The elite counter-terrorism service (CTS) had taken full control of the Baladiyat district and encircled neighboring Sukkar, Sabah al-Numan told Reuters.
The advance also consolidated Iraqi forces' control of several districts close to the ruins of the ancient Assyrian city of Nineveh, east of the river.
An offensive to drive Daesh out of Mosul, its last major Iraqi stronghold, has gained momentum since the beginning of the year.
CTS forces reached the east bank of the Tigris Sunday, an advance which will eventually enable them to begin assaults on the city's west, all of which Daesh still holds.
Recapture of Mosul after more than two years of Daesh rule would probably spell the end of its self-declared caliphate, which spans areas of Iraq and Syria.