Rutherford, 26, will play in all three forms of the game while Dilshan plays in the Caribbean Premier League.
Rutherford, who is part of the Kiwi Test side touring England this summer, has played 16 Tests, scoring 755 runs.
"Hamish is a talented top-order batsman," said elite performance director Graeme Welch.
"He has international pedigree in both the long and shorter formats which will be key as he joins us at what is a busy part of the summer."
Rutherford also had a spell playing for Essex during the 2013 season.
Derbyshire have signed New Zealand opening Test batsman Hamish Rutherford as cover for Tillakaratne Dilshan from mid-June until the end of July.