The work will be carried out by Network Rail as part of its electrification programme across the north.
Network Rail will now develop a detailed plan for the work, which could be completed as early as 2017.
Transport Minister Baroness Kramer will confirm the route will form part of a wider regional electrified route reaching Kendal, Burneside, Staveley and Manchester.
She said: “Electrifying this key rail link will support the vital tourism industry in the area, and help us build a stronger economy in the Lake District and beyond.
Other routes being electrified include Manchester to Liverpool, Preston, Leeds and York, and Preston to Blackpool.