In a statement, the group said it backed a US plan for air strikes in Syria and training of rebel forces, but that a "stable and extremist-free region" required "degrading and ultimately removing the Assad repressive regime."
Earlier, US President Barack Obama pledged a "relentless" war against the Islamic State, a jihadist group that has declared an Islamic caliphate in large tracts of territory in Syria and Iraq.
The group has carried out abuses including beheadings and crucifixions, and faced a backlash from Syrian rebel groups opposed to its violations and harsh interpretation of Islam.
Obama said he would "not hesitate to take action against ISIL in Syria," using a different acronym for the group, and pledged to strengthen Syria's opposition.
The Syrian Coalition said it had "long called for this action, and... warned time and again of the growing threat of this extremist group."
"We urge the US Congress to approve the president's policy as soon as possible, and allow the training and equipping of the Free Syrian Army," it said.
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The moderate Free Syrian Army "can succeed, but it needs the necessary support that would enable it to form a reliable and well-equipped force," the statement added.
It warned however that it was key to "realise that the Assad regime represents the root cause of the violence, brutality and sense of impunity prevailing in Syria."
Combating the Islamic State "alone cannot bring about a stable and extremist-free region," the group said.
"It takes degrading and ultimately removing the Assad repressive regime seeking the perpetual destabilisation of the entire region for the sole purpose of staying in power."
Syria's government has sought to present itself as a partner in the fight against jihadists including the Islamic State.
But it has insisted any military action in its territory must be done in coordination with the government in Damascus.