The safe passage leads to the southwest of Fallujah, along a road where the Islamic State group killed at least 18 members of two families trying to flee on Friday.
"The army opened a safe corridor for families fleeing from Fallujah through Al-Salam intersection," an officer with the Joint Operations Command supervising the fight against IS said.
Around 24,000 people have fled IS rule since Iraqi forces launched an offensive on Fallujah on May 22-23, but very few have been able to leave central Fallujah where the jihadists are using civilians as human shields.
"The latest figure we have is that 4,000 individuals have managed to get out over the past 24 hours," the NRC's regional media adviser Karl Schembri said.
"We are of course relieved, but it also means we are completely overwhelmed as a humanitarian community," he told AFP, warning that the available resources of safe drinking water would not meet the needs of all the displaced for much longer.
The NRC's Schembri said that the general aid effort in Iraq was massively underfunded, hampering the delivery of urgent relief.
In the short term, he said, the response to the Fallujah operation would require $10 million (8.9 million euros) over the next six months if another 35,000 people were displaced.
Before the safe passage was opened on Saturday, the NRC estimated that 50,000 civilians were trapped inside Fallujah.
- IS killed 18 -
Residents had been taking huge risks to flee, walking along mined roads and trying to cross the Euphrates River at any cost.
In the worst known incident so far, 18 people were killed by IS as they approached the Al-Salam intersection on Friday.
"A number of residents were trying to flee and as they neared the Al-Salam intersection, Daesh (IS) opened fire on them, killing 18 and wounding dozens," a senior officer at the Joint Operations Command told AFP.
The officer, speaking on condition of anonymity because he is not authorised to talk to the press, said the army was able to rescue some of the wounded.
Relatives said the initial group that tried to sneak out of IS-controlled areas on Friday included around 100 people, the majority being women and children.
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The group were all from the same two families -- Albu Hatem and Albu Saleh -- and had made a previous failed attempt to leave days earlier.
"When they got near the intersection, which is the meeting point with the Iraqi forces, two gunmen on motorbikes arrived and sprayed them with gunfire," said Ahmed al-Ghneim, a relative.
- Steady progress -
Two survivors, relatives from the Albu Saleh family, are staying at his home in Amriyat al-Fallujah, south of Fallujah.
"Some of the residents jumped into the canal, some fled to a nearby house. When they entered it, it blew up on them because it was booby-trapped," he said.
"Some survivors were forced to go back inside Fallujah. Daesh took 17 of the wounded to Fallujah hospital," Ghneim said.
Sami Abu Hatem, a relative who was already living in a camp in Amriyat al-Fallujah, confirmed that version of events.
"Three of my direct relatives, a man with two young children, were among those massacred," he said.
Abu Hatem said he knew of 18 members of the group being killed and added that more were feared missing after jumping into the canal.
"This is a major crime, it's a massacre, entire families are being eliminated," he said.
Ghneim said IS smugglers would charge $100 per person as guides to avoid bombs and booby traps.
"My family walked like that with all the other people for six kilometres (3.5 miles)," he said.
NRC and the other groups organising the relief effort around Fallujah said they expected more civilians to flee in the coming days.
Iraqi elite forces have been making slow but steady progress in the city centre in recent days.
On Sunday, they retook another district in central Fallujah that brings them within three kilometres of the main government compound, Lieutenant General Abdelwahab al-Saadi, the operation's overall commander, told AFP.
Also Sunday, Iraqi forces backed by US-led coalition air strikes advanced towards Qayyarah south of the main IS hub of Mosul, reviving a weeks-old operation that has made slow progress so far.