"And with this ... incident that has caused so much damage would be resolved," Correa said before standing up and shaking hands with Uribe.
Trade to resume
Chavez later announced that Venezuela would re-open its Colombia border to trade, agencies said.
Separately, Daniel Ortega, Nicaragua's president, said that his country would re-establish diplomatic ties with Colombia, a day after breaking them off.
The developments came on the same day that Bogota announced that a second senior commander of Farc (Revolutionary Armed Forces of Colombia) had been killed in western Colombia by his own men.
The Dominican Republic summit was organised by the Rio Group, comprising 20 Latin American democracies, created in 1986 in part to find solutions to regional problems.
The diplomatic spat began last weekend when Colombian forces crossed into Ecuadorean territory and killed Raul Reyes, a senior Farc commander, and several other fighters.
Venezuela and Ecuador broke off diplomatic relations with Colombia and sent troops to their Colombian borders.
|Ecuador and Venezuela both ordered troops|
to their borders during the crisis [EPA]
Uribe had accused Chavez, who had helped broker several hostage releases by Farc prior to the raid, of aiding the rebels.
At one point, Uribe had said he would bring charges against Chavez at the International Criminal Court.
At Friday's meeting, Uribe and Correa traded insults, with Uribe saying Correa had not co-operated in "the fight against terrorism".
Uribe alleged that Colombian forces had found a letter in the Ecuador raid which mentioned "aid delivered to Rafael Correa, as instructed".
Correa responded by calling Uribe a liar, saying "your insolence is doing more damage to the Ecuadorean people than your murderous bombs.
"Stop trying to justify the unjustifiable."
Al Jazeera's correspondent Lucia Newman said the handshakes followed a tense morning of insults and accusations which eased as other Latin American leaders called for calm and reflection and warned that the stability of the region was at stake.
However, the real problem has not been resolved, she said: Colombia shares a border with five other countries and as long as rebels and paramilitary groups continue to seek refuge in neighbouring countries, there will still be problems.
While Ecuador has accepted apology for the raid, Colombia has not committed to never carrying out such an act. Therefore it remains a fragile peace, Newman said.
The diplomatic drama unfolded as Juan Manuel Santos, the Colombian defence minister, identified the second slain Farc commander as Ivan Rios.
Rios was reportedly shot dead by his own chief of security, who offered his severed hand as proof to the Colombian authorities, Santos said in Bogota on Friday.
"Farc has suffered a new, major blow," he said.
|Rios was originally thought to have been killed|
in a Colombian military raid [AFP]
Rios was believed to be the youngest member of Farc's secretariat and considered one of the closest aides to Manuel Marulanda, the group's founder.
Initial reports said Rios had been killed in a Colombian military raid.
The motive for the killing is unclear, although the Reuters news agency reported that Colombia has increased the rewards it pays to those who help capture or kill rebel fighters.
In a related development, Farc has provided "proof of life" of 10 more hostages in the run-up to a possible new release.
However, Ingrid Betancourt, the French-Colombian politician, is not one of them, Ramon Rodriguez, the Venezuelan interior minister, said on Friday.
The hostages are thought to be Colombian soldiers.