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.

Police in Tamaulipas state said the hole, which had been hidden by bricks, was up to five metres (16ft) deep.
They said it was unfinished and no one has escaped.
In 2015, one of the country's biggest drug lords, Joaquin "El Chapo" Guzman, escaped via tunnel from Altiplano prison.
He was recaptured after six months on the run and later extradited to the US.
The latest hole was in a jail in Reynosa, a city close to the US border.
It was found after authorities used geothermal technology to detect activities underground, according to local media.
In March, 29 inmates escaped via a tunnel from a prison in the Tamaulipas state capital Ciudad Victoria.
One of the prisoners who fled was involved in the 2012 kidnapping and murder of Karen Alejandra Rodríguez.
He was later recaptured, but the victim's mother, Miriam, who had been campaigning for justice for Mexico's missing people, said she lived in increased fear after the escape.
She was shot dead in her home last week by an unknown assailant.
Mexico also saw another major prison break in the country's north-west in March.
Juan José Esparragoza Monzón, son of one of the country's other drug kingpins, escaped from a jail in Culiacán, Sinaloa state, alongside four other inmates.
His father, Juan José Esparragoza, was a long-time partner of El Chapo in the Sinaloa Cartel, one of the country's biggest crime syndicates.
Monzón remains on the loose.
Tamaulipas police said on Sunday that they were conducting a search for further holes after the latest tunnel was discovered on Saturday night.
They said that Reynosa prison was operating as normal and visiting hours had resumed.
The loss of 1,000 jobs at Middlesbrough Council over the next three years is the "best case scenario", mayor Ray Mallon has said.
[NEXT_CONCEPT]
A tunnel has been discovered in a Mexican prison, stashed with drugs, alcohol and sharp tools, according to local authorities.