Up to 1,500 staff could be axed by 2017 due to a funding gap of almost Â£70m.
Mr Mallon said services would be closed and multimillion-pound projects reviewed in an attempt to stop the town "fraying at the edges".
The government said Middlesbrough received more funding than the national average and it should cut wastage.
Mr Mallon outlined the financial situation to councillors at a meeting on Wednesday and said he hoped action taken would "reduce substantially" the number of potential job cuts, but the loss of about 1,000 posts was "the best case scenario".
He said it was a "real shame" staff would be made redundant but insisted the council was "duty bound" to cut spending.
The council has just over 2,500 full-time staff, excluding those in schools.
More than Â£32m of the required savings are necessary due to changes in government funding and the authority is facing an increased demand on care services, Mr Mallon said.
"By 2018 or 2019, 75% of the council's budget will be spent on helping children and the elderly.
"The temperature has been rising for a long time and it's still rising."
Ahead of the meeting, Labour group leader Charlie Rooney said the town was being "devastated" by government cuts.
Conservative group leader Chris Hobson called on the authority to "start doing things differently".
"We've got to start running this council as a business," she said. "We've got to start bringing money in ourselves and thinking about how we can do that."
Mr Mallon will announce a list of specific budget proposals next month.
The loss of 1,000 jobs at Middlesbrough Council over the next three years is the "best case scenario", mayor Ray Mallon has said.