Rather than closing today, Anlaby Park Library, in west Hull, will stay open until the end of May to give residents time to come up with proposals to run it as a community venture.
Council leader Steve Brady said: “Listening to the concerns of the community and the representations from the Hull West and Hessle MP Alan Johnson, helped me to make this decision.
“Since the government’s budget announcement last week we now know that there will be with further cuts to local government and we are expected to find a further £6.2m worth of savings.
“This means we have to find savings of £20.6m in 2014/15 as well as ensuring that our plans for a balanced budget for 2013/14 are achieved.
“Tough decisions will continue to be made but where possible we will listen to our local communities.”
Barbara Bourne, who runs a free craft group on Tuesdays, and presented a 2,000-name petition to councillors, calling for the library to stay open, said: “It does give us breathing space and that is something.
“At least we have some time to find a way to keep the library open.”
But Coun Claire Thomas, Lib Dem chair of West Area committee, said the whole process had been “farcical.”
She said: “The initial decision was deeply flawed and Labour has failed to properly take on board the concerns of residents. The council leader’s concession of a stay of execution for two months simply doesn’t go far enough.
“Labour Councillors are still refusing to listen to local residents, and their announcement today does not commit them to carry out a consultation which is what local residents are calling for.
“Last night I met local residents who want to save the library.
“The message from that group was clear; they want the library to stay open.”