Eritrea has repeatedly denied the allegation.
Diplomats said Uganda, which has peacekeeping troops in Somalia that have been targeted by al-Shabaab, drafted the resolution after the African Union called on the council in May to punish Eritrea over its role in Somalia.
But Eritrea charged that its true author was the United States.
Araya Desta, Eritrea's UN Ambassador, described the resolution as "shameful" and told reporters it was based "on fabricated lies, mainly concocted by the Ethiopian regime and the US administration".
Ethiopia, Eritrea's regional rival, invaded Somalia in 2006 with tacit US backing to rout an Islamic Courts movement from Mogadishu. It withdrew its troops earlier this year.
"We have never supported any insurgents or any opposition in Somalia," Desta said.
"We don't want to take sides in Somalia. Somalis are our brothers."
But a UN arms monitoring body, set up to record violations of the 1992 arms embargo on Somalia, has said Eritrea was sending munitions and giving logistical support to Somali rebels.
The measure demands that Asmara "cease all efforts to destabilise or overthrow, directly or indirectly" the Transitional Federal Government (TFG) in Somalia.
It urges member states to conduct inspections on their territory, including seaports and airports, of "all cargo to and from Somalia and Eritrea" if there is reasonable grounds to believe the shipments contain banned weapons or related material.
The text also presses Eritrea to withdraw troops immediately from disputed territories along its frontier with Djibouti, and engage in diplomatic efforts leading to "a mutually acceptable" settlement of their long-running border dispute.
Eritrea's dispute with Djibouti dates back to June 2008, when the two countries clashed after Djibouti accused Asmara of moving troops across the border. The Security Council says Eritrea has ignored its demands to withdraw the troops.