A Dagestani security official said on Tuesday that the rebels had released all their hostages before fleeing to an undisclosed location.
And Russian Defense Minister Sergei Ivanov announced that troops had shoot-to-kill orders if they came across the men.
Ivanov said Russian troops, who are estimated to number around 700, had received orders to "arrest or eliminate" the guerrillas on site.
The hostage drama began early on Monday when several dozen fighters killed nine border guards in an apparent ambush near the Chechen border with Dagestan.
The security official said the rebels took four people hostage and eventually fled to the western Dagestani village of Galatli, where they took another seven people hostage before being surrounded.
The rebels released all their hostages before fleeing on Monday evening.
"They (the Chechens) never planned to hurt us"
A senior Dagestani official quotes one of the hostages
But one official said they never threatened their hostages or made any political demands.
"They never planned to hurt us," Magomed Abullayev, a senior official in the Dagestani interior ministry, quoted one of the hostages, Patimat Ibragimova, as saying.
She said up to 40 guerrillas were involved in the raid.
"Most of them were Chechens - I could tell by their accent," she said, before adding that some of the rebels appeared to have come from abroad.
The incursion has led Russian news for two days, fanning fears that Chechen resistance to Russian occupation is growing.
Some reports suggest the initial group that entered Dagestan has since split up into at least three.
Several were reported to be carrying hostages with them and the largest group apparently entered Galatli.
Officials said on Monday that a state of emergency had been declared in the Dagestan region.
Putin lauched the Chechen war in
The incident revived memories of August 1999, when several hundred separatist Chechens stormed into Dagestan and took over two villages in a region more than 100 km north of the latest fighting.
The incursion lasted several weeks and sparked the second Russo-Chechen war.
The then prime minister, Vladimir Putin, sent Russian troops into Chechnya on 1 October 1999 for what was supposed to be a lightning-strike operation to root out "terrorists".
However, the Chechen war has dragged on and has settled into a guerrilla conflict that continues to claim civilian, troop and rebel lives on a nearly daily basis.
Despite the Kremlin's efforts to portray the war as over and the situation in Chechnya as returning to normal, an estimated 2000 rebels continue to hide out in the mountains hugging the southern border of the republic.