The injured trio of Justin Langer (concussion), Shaun Tait and Michael Kasprowicz (both with back injuries) will be rested after helping Australia to a 3-0 Test series whitewash in South Africa.
They will be replaced by veteran quick Gillespie, New South Wales batsman Phil Jaques and Queensland fast bowler Mitchell Johnson.
Gillespie, 30, a cult favourite among Australian fans because of his extravagant mullet haircut, was dropped mid-way through Australia's Ashes loss to England last year and has fought hard to prove he retains the zip that earned him 69 Test caps.
He was considered unlucky to miss out to fellow Ashes discard Kasprowicz on the South African tour but outgoing chairman of selectors Trevor Hohns said his work for South Australia in the domestic competition earned him a Test recall.
Jaques, 26, receives a second chance after making only one Test appearance for Australia in last year's Boxing Day Test against South Africa in Melbourne, when he scored a total of 30 runs in two innings.
Test debutante Johnson, 24, will be seeking to improve on his performances in his two one-day internationals, where he has taken no wickets and conceded 92 runs.
Hohns said selectors were forced to make changes to replace the injury-hit trio before the two-Test tour of Bangladesh begins in Fatullah on Sunday.
"Our advice was that the players would be definitely unavailable for the first Test and maybe the second too," he said.
"Given that it is only a two-Test series, we thought it would be in the best interests of the players concerned to give them a full recovery. We also felt it was important to have a squad in Bangladesh who were all available for the first Test.
"The three players who are coming into the squad have performed very well in domestic cricket and deserve their opportunity to tour."