Dilshan, who did not travel with the team to India, is expected to be available for the second match, in Ranchi, on February 12.
The first match is scheduled to be held in pune on February 9, with the third game at Visakhapatnam on February 14.
Dilshan's injury is an aggravation of an old complaint that surfaced after he hurt himself while attempting a catch during the semi-final of the recently-concluded Super T20 Provincial Tournament.
He is likely to resume training in the nets on Monday.
Dickwella earned his call-up following a good run in the Super T20 tournament, where he struck 189 runs in six innings, with a strike rate of 173. He has played four Tests and one ODI for Sri Lanka, but is uncapped in the shortest format.