"This is a revolution for freedom, for dignity and against dictatorship," he added.
El-Bahra was earlier elected president of the coalition with 62 votes. His nearest rival, Mowafaq Nayrabiyeh, won 41 votes, while Walid el-Omari won just three votes, it added.
Delegates also Wednesday elected Naser al-Hariri as secretary general as well as three vice presidents.
In a first for the coalition, one of the vice presidents elected was a woman, Noura al-Ameer.
'The world has given up'
El-Bahra will have the task of keeping alive the campaign to unseat Assad amid territorial gains by the regime and the rise of the radical jihadist group Islamic State, which the coalition vehemently opposes.
Jarba headed the coalition from July 2013 but failed in efforts to unite the opposition and obtain significant Western military support.
El-Bahra was born in Damascus in 1959 and studied industrial engineering in the United States, according to the coalition's website. He speaks English fluently.
But he has spent most of his adult life in Saudi Arabia, where he has managed several hospitals and businesses and still lives.
He headed the opposition negotiating team in its delegation to the failed Geneva 2 talks between the opposition and the regime in Switzerland earlier this year.
El-Bahra is softly-spoken yet he has frequently publicly denounced human rights violations by the Assad regime, as well as the international community's failure to step up assistance to the Syrian people.
"In reality it is impossible for the international institutions and groups to stick to their promises," he told reporters.
"They have given up and can no longer satisfy the real needs of the situation in Syria," he added.
He described the crisis in Syria, which has left more than 162,000 dead since March 2011, as a "humanitarian and political crisis without equal that has passed all limits".
"It is one of the biggest crises in history and a terrible tragedy."
The Syrian opposition has been riven by internal conflicts linked to disputes between its main foreign sponsors, notably Saudi Arabia and its influential Gulf Arab neighbour Qatar.
But its members are now trying to reach a consensus and end the feuding which has exasperated the West and anti-Assad Syrians tired of being manipulated by outside powers.
As a Saudi based businessman, el-Bahra should enjoy the support of Riyadh but will have to show he can also be a figure of reconciliation within the opposition.
UK Foreign Secretary William Hague welcomed el-Bahra's election and pledged that Britain would work closely with him and others in the SNC.
The president of the coalition changes on an annual basis partly in an effort to ensure no foreign power wields too much influence by keeping a pliable figure in power too long.
Most Syrians living in opposition-held areas believe the coalition has failed to secure either the political or military backing the rebels need to topple Assad regime, with many activists frequently accusing its members of being disconnected from reality on the ground.