It brings the total amount promised by the authority to more than £400,000 out of a total cost for the project of nearly £5 million.
Shona Alexander, the council's director of regeneration, told a meeting of the full council that finance for the project was coming from a number of areas.
She said £3.169 million had already been secured from One NorthEast, the Heritage Lottery Fund, Youth Hostel Association, the borough council, English Heritage and private trusts.
However, that still left almost £1.8 million to be found.
A bid for £1 million has been made to the Sea Change fund, a national scheme run by the Commission for Architecture and the Built Environment, and One NorthEast had agreed to cover the rest, provided both Berwick Borough and Northumberland County councils ploughed in an extra £100,000 each.
Mrs Alexander said she was confident the £1million Sea Change bid would be successful and urged councillors to agree the £100,000.
Under the proposals, Dewar's Lane Granary will provide a ground floor bistro, with facilities for offices and seminars on the first floor. The second floor will have exhibition space while the third and fourth floors will be turned into a youth hostel.
If the project progresses, the lower floors will be operated by the Berwick Preservation Trust, with the Youth Hostel Association running the top floors.
It is estimated the refurbished granary could attract 22,000 visitors a year.
Council leader, Coun Isabel Hunter, urged members to support pledging the extra £100,000 to the project.
"If we decide not to go ahead with this, the money would go to the unitary council and would be lost to Berwick borough."
Coun Dougie Watkin added: "This is the last chance saloon. Given the time factor, this is the last opportunity this council will have to sort out the granary.
"We are being asked to promise £100,000 that will bring almost £5 million of investment to Berwick.
"It is £5 million that will promote revenue year after year. This is our chance to do something about."
However, support for providing extra cash for the granary was not universal.
Coun David Stewart questioned exactly where the extra £100,000 would come from, and was told it would be raised through the sale of "certain council assets".
Coun Stewart said: "The £100,000 has a direct link to the sale of the council car park, but I seem to remember we had a debate on whether the car park at the Quayside was suitable for sale and whether we should sell to One NorthEast."
He said councillors may wish to ask how money could be found for the granary project but not to address other issues around the town.
"We are now told our contribution is more than £400,000 and it magically appears. In my view, it is little wonder that residents of Berwick hold this council in low esteem."
Coun John Robertson said a better legacy for the council would be to spend the money promised to the granary in other areas, such as social registered housing or on a replacement car park.
"All in all, I think we are being driven down one road using proceeds from the sale of land which was questionable at the time and I will not be voting in favour of it."
Coun Brian Douglas was another who thought the money could be better used elsewhere and described the granary project as a "white elephant".
But Coun Peter Watts countered: "If it comes off, we are being asked to put in just eight per cent of a £5 million development. I think we ought to go for it."
Mayor, Coun Hazel Bettison, added: "If we do not use this money, it is only going to go to the unitary council.
"You had your chance to put other ideas forward, where were they? We have only got six months left as a council and there is no time to put another project together."
Coun Diana Beith said: "I don't think the people of Berwick would thank us for saying no to investment of £5 million."
She added that if the extra £100,000 was not pledged, and the project fell as a result, it would cost the council much more to deal with the granary remaining derelict.
Spelling out the situation, Mrs Alexander explained: "If we don't get this funding package together, then all the money is denied to this project and we will be left with a disused liability.
"If we don't do it, we will be handing over the keys of a liability to the new authority."
After all members who wished to speak on the issue had done so, Coun Bettison asked for a vote to be taken on the recommendation that the council provides a further £100,000 to the project, and it was carried 17-5.