Ashraful will be replaced by middle-order batsman and leg-spinner Alok Kapali for Wednesday's match in Fatullah, near Dhaka, chairman of selectors Faruq Ahmed told AFP.
"Ashraful has had a lean patch for the last few matches. We thought he needed rest for one or two games," said Ahmed.
"There is no doubt he is a very good batsman. But there are also too many expectations from him, which I think have become a burden for him."
Ashraful, 21, scored five runs from 23 balls in Sunday's first One-dayer in Chittagong which world champions Australia won by four wickets.
Ashraful, whose century led Bangladesh to a memorable One-day victory against Australia in England last year, also failed to get past 50 during his team's 4-0 sweep against Kenya last month.
Kapali, 22, was a last minute exclusion from the Chittagong match after being struck on the face by a ball during a practice session.
Seamer Shahadat Hossain was also included in place of Syed Rasel for Wednesday's game.
Bangladesh 12: Habibul Bashar (capt), Khaled Mashud, Javed Omar, Mohammad Rafique, Rajin Saleh, Mashrafee Bin Mortaza, Aftab Ahmed, Tushar Imran, Abdur Razzak, Alok Kapali, Shahriar Nafees and Shahadat Hossain.