Bird's call-up came hours after Queensland leg-spinner Mitchell Swepson was added to the squad as a replacement for injured fast bowler Mitchell Starc.
Pattinson was suffering inflammation around an area of his back where he had a previous stress fracture.
Cricket Australia said on their website the injury was not another stress fracture and the decision to withdraw him from the tour was made to ensure he was fit for the Ashes series against England later this year.
Uncapped Swepson, who was previously selected for the tour of India earlier this year but did not play, was selected instead of a pace bowler because of the expected spin-friendly conditions in Bangladesh.
Swepson is the third frontline spinner in the side, with off-spinner Nathan Lyons and left-arm orthodox spinner Ashton Agar also in the squad.
"We are comfortable with the fast-bowling stocks we have in the squad so have opted to add an additional spinner given the conditions we are likely to face in Bangladesh," selector Trevor Hohns said in an earlier statement.
"Mitchell is a very exciting young leg-spinner who we think will benefit immensely from further experience in the sub-continent."
Starc suffered a stress fracture in his foot on the tour of India and was sent home. He returned for the one-day Champions Trophy tournament in England in June but later felt the foot had not healed properly.
The original squad for the two-test tour was named in June and selectors had expected to use Australia's 'A' tour of South Africa last month to decide on a possible fast bowling replacement for Starc if he was ruled out.
That tour, however, was cancelled due to a long-running pay dispute between Cricket Australia and the players which was only settled this week.
The first test is in Dhaka from Aug. 27-31 and the second in Chittagong from Sept. 4-8.
Australia test squad: Steve Smith (captain), David Warner, Ashton Agar, Jackson Bird, Hilton Cartwright, Peter Handscomb, Josh Hazlewood, Usman Khawaja, Nathan Lyon, Glenn Maxwell, James Pattinson, Matthew Renshaw, Mitchell Swepson, Matthew Wade.
(Reporting by Greg Stutchbury in Wellington; Editing by Peter Rutherford)
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