A delegation spoke to Belfast councillors as dozens of their colleagues watched from the public gallery in City Hall.
Workers have occupied the yard since Monday, holding a 24/7 picket as the clock runs down towards the arrival of administrators at the start of next week.
Harland and Wolff, birthplace of the ill-fated Titanic and a business synonymous with Belfast's industrial heyday, has been unable to find a buyer after being put up for sale amid serious financial problems at its Norwegian parent company.
The council convened for a special meeting on Friday afternoon to debate a motion calling for intervention to help stave off closure.
In an emotional speech, steel worker and union representative Joe Passmore said the city's proud shipbuilding tradition faced extinction.
"This is my city and I believe in it very strong, and I have a great affiliation and I am very proud to be from here, but I don't want to see our greatest industry, that iconic industry, going down the drain," he told councillors at the start of the meeting.
"I am not prepared to let that happen and all my fellow trade unions are not prepared to let it happen.
"What we don't want you to tell us, we don't want you to tell us what you can't do, because we have had that all week.
"I don't want you to tell us 'tea and biscuits and there, there'. I have had that all week and I am sick of listening to it.
"I want to know what you can do to help because we are on our last legs right now. We don't have much time to go before all these people go on the dole. When we lose these skills how are we ever going to get them back?
"When that closes and they build a marina down there, how is industry ever going to come back into Belfast? It's the beginning of the end.
"Let Belfast take the lead on this, let Belfast take the lead for the whole of the United Kingdom and let us concentrate on our industry."
Councillors applauded Mr Passmore at the close of his speech, as they did after each workforce representative addressed them.
Workers held a rally outside City Hall at Friday lunchtime ahead of the meeting.
The emergency session was called by SDLP councillor Brian Heading and Green Party councillor Anthony Flynn.
They tabled a motion which would see the council convene an urgent forum between trade unions, the state-funded business support agency Invest NI, Stormont's Department for the Economy, and the UK Government to secure the future of the shipyard.
Harland and Wolff, whose famous yellow cranes Samson and Goliath dominate the Belfast skyline, once employed tens of thousands but currently has a workforce of around 130.
It has diversified away from shipbuilding in the last two decades and now primarily works on wind energy and marine engineering projects.
The Government has so far declined to intervene to bail out the yard, potentially in the form of nationalisation, insisting it is a commercial issue.