WASHINGTON — The Obama administration, anxious to showcase a success in the fight against the Islamic State, sent out a flurry of statements and news releases on Monday hailing the Iraqi military’s take-back of Ramadi from the militant Sunni extremist group.

But the accompanying military campaign must now move quickly to capitalize on the success, Defense Department officials said, so the American-led coalition that is fighting the Islamic State does not lose crucial momentum.

That means the Iraqi military must finish clearing pockets of Islamic State resistance out of Ramadi, the capital of Anbar Province, turn it over to a holding force of Sunni tribal fighters and move swiftly to the next stages in the war. Those include pushing Islamic State militants out of the Euphrates River valley and the Tigris River valley north of Baiji; retaking the next Anbar Province city under occupation, Falluja; and finally turning to Mosul, whose liberation is likely to be the last piece of the puzzle to defeating the Sunni militancy, at least in Iraq, defense officials said.

And that, administration officials acknowledge, will take months, if not years.

“Don’t forget, the enemy gets a vote, too,” Col. Steven Warren, the American military’s spokesman in Baghdad, said on Tuesday. “This is all about condition setting. We have to be able to have enough territory under our control to create supply lines, and mutually supporting staging areas, to get into Mosul.”