Gandhi also expressed the hope that he would work to further improve bilateral relations.
He hoped that India and Pakistan will be able to resolve their outstanding issues during Sharif's tenure.
Sharif was quoted by a PML-N spokesman as saying that all outstanding issues with India would be resolved through dialogue.
The resolution of problems between the two neighbours is vital for ensuring sustainable social economic development of the people of the two nations, Sharif said.
Earlier, Prime Minister Manmohan Singh had congratulated Sharif and invited him to visit India.
Sharif also said that he would be "very happy" to invite Singh to Pakistan for his oath-taking ceremony as the new Premier.
Sharif is poised to take over as Pakistan Prime Minister for a record third time after his party emerged victorious in the May 11 general elections
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