As expected, the selectors opted to rest England captain Strauss, with vice-captain Cook promoted to skipper for the two-Test and three one-day international tour to Bangladesh.
Paul Collingwood will captain the side in the two Twenty20 internationals in Dubai which begin the tour.
James Anderson has also been rested from the tour in order to undergo a specialist review and a programme of rehabilitation for his chronic right knee injury.
It is anticipated that following the rehabilitation period, the Lancashire seamer will be fully fit and available for selection for the World T20 tournament in the West Indies in late April.
Yorkshire pace bowler Shahzad - who recently signed a three-year contract at Headingly Carnegie - is one of two uncapped Test players, alongside Kent off-spinner James Tredwell, have won places in both squads, while another uncapped player Hampshire batsman Michael Carberry has been included in the Test squad.
National selector Geoff Miller believes it was important Strauss was given the opportunity to recuperate from the 10-week tour of South Africa ahead of a gruelling summer and next winter's Ashes campaign in Australia.
Miller said: "Andrew Strauss has provided outstanding leadership for the team in both forms of the game over the past 12 months and the selectors feel it is important that he takes a break ahead of an extremely busy programme of international cricket leading up to and including the Ashes series in Australia and the ICC Cricket World Cup in 2011."
Miller insists England will be in safe hands and is confident in the squad's leadership under Cook and Collingwood.
He added: "Our decision to appoint Alastair Cook to the Test vice captaincy last year clearly demonstrated the belief that he has the potential to be a future England captain.
"Alastair will now have an opportunity to develop his leadership skills still further by leading the side in both forms of cricket in Bangladesh and I know that he is excited by the challenge and looking forward to working closely with Andy Flower.
"Paul Collingwood had an excellent tour of South Africa in all forms of cricket and he will continue to lead the side in the Twenty20 format with the two Twenty20 internationals against the current world champions Pakistan providing excellent preparation for us ahead of the World Twenty20 in the Caribbean."
Uncapped trio Shahzad, Tredwell and Carberry are players who have impressed the selectors, with Miller tipping each of them to shine on the forthcoming tour.
"We are delighted to welcome Ajmal Shahzad to both squads for the first time," added Miller.
"He enjoyed an excellent season with Yorkshire last year, made a strong impression while he was in South Africa with the EPP squad and is a bowler with the potential to make a real impact in international cricket.
"James Tredwell and Michael Carberry have also continued to impress the selectors over the course of the winter and we expect both players to make a strong challenge for inclusion in our starting line-up on the forthcoming tour."
Yorkshire's Tim Bresnan has also been called up to the one-day squad for the tour.
England squads for tour of Bangladesh:
One-day international squad: A Cook (captain, Essex), T Bresnan (Yorkshire), S Broad (Nottinghamshire), P Collingwood (Durham), J Denly (Kent), E Morgan (Middlesex), M Prior (Sussex), K Pietersen (Hampshire), L Plunkett (Durham), R Sidebottom (Nottinghamshire), A Shahzad (Yorkshire), G Swann (Nottinghamshire), J Tredwell (Kent), J Trott (Warwickshire), L Wright (Sussex).
Test squad: A Cook (captain, Essex), I Bell (Warwickshire), S Broad (Nottinghamshire), M Carberry (Hampshire), P Collingwood (Durham), S Davies (Surrey), G Onions (Durham), K Pietersen (Hampshire), L Plunkett (Durham), M Prior (Sussex), A Shahzad (Yorkshire), R Sidebottom (Nottinghamshire), G Swann (Nottinghamshire), J Tredwell (Kent), J Trott (Warwickshire), L Wright (Sussex).