Mr Corbyn received warm applause from Labour candidates who were waiting to find out the results of the Sheffield City Council elections.
Counting of votes for the 84 council seats began at 9.30am this morning, with the first results expected at about midday.
Labour is looking to retain control of the council, which it held with 59 seats before yesterday’s election, to the Lib Dems’ 17, the Greens’ four and UKIP’s four.
In her victory speech, Ms Furniss said: “ Tonight the people of Sheffield Brightside and Hillsborough have sent David Cameron an emphatic message.
“Mr Cameron, we have had enough of your uncaring government.
“We’ve had enough of your Tory government’s unfair cuts, which have hit communities like ours in Sheffield so much harder than more affluent areas.
“We’ve had enough of your Tory government giving tax cuts to the rich while taking money away from the disabled and vulnerable.
“And we’ve had enough of your Tory government dragging your feet on help for our steel industry.
“I want to thank all the thousands of people who voted for me to send me down to Westminster as their MP. Thank you for putting your faith in me, and in the Labour Party, to fight for you in Westminster.
“And to those who voted for other candidates or didn’t vote at all – I promise I will be your strong advocate in parliament too; I’ll always do what’s right for the people of my home city.
“This was a by-election none of us wanted. My dear husband Harry was a fantastic MP for this constituency.
“He was also a loving husband and father, and we all miss him a great deal.
“His time in parliament was short but he fought with determination to do what was right for local people, right to the end.
“He would have wanted me to stand in this election, to carry on his work – both here in Sheffield and in Westminster;
- To work for better opportunities for young people.
- To stand up for those on low incomes and vulnerable people.
- And to fight for a fair deal for Sheffield against this out of touch and cruel Tory government.
“My commitment is to hold this government to account for their failures on the health service.
“To fight for more resources for our local schools and colleges.
“And to fight for more homes people can afford...
“Above all, I’ll fight to save our steel industry – the jewel in Sheffield’s crown and in doing so I’ll remember Harry’s last words in the House of Commons to David Cameron – enough of the hand-wringing and crocodile tears – it’s time for action.
“Thank you to the returning officer and your team for overseeing a well-organised and orderly contest and count.
“Thank you to the police for your role in ensuring a safe and calm election.
“Thank you to my fellow candidates for the positive manner in which the campaign has been fought.
“And above all Thank you to my agent and campaign team, and to the hundreds of Labour members, from Sheffield and beyond, who came and campaigned for us without your help we wouldn’t have won this fantastic victory.
“For me, the hard work starts now – I promise I’ll be a strong voice for you in parliament.”