Batsman Martin Guptill (finger), all rounder Daniel Vettori (Achilles) and swing bowler Tim Southee (ankle) were not considered for the series, which starts in Chittagong on October 9.
Southee and Guptill were in New Zealand's Test squad for their 2-0 series loss in England earlier this year, while former New Zealand captain Vettori has had surgery on the injury that has kept him out of most cricket for the best part of a year.
"Essentially this squad builds on the team that toured England in May," new national selection manager Bruce Edgar said. "The key was to take a consistent approach to selection and provide continuity for the players."
Sodhi and Anderson are both with New Zealand 'A' in India and Anderson scored a century earlier this week against India 'A'. He played Twenty20 and one-day cricket for New Zealand earlier this year.
Sodhi has progressed into the squad after good performances for age-grade New Zealand sides and coach Mike Hesson said Bangladesh would be a good place to give the leg spinner an opportunity to continue learning his craft.
"The Bangladesh tour is a good opportunity to introduce a promising spinner, and having two specialist spinners in the squad gives us the option to utilise wickets that are likely to turn," Hesson said.
"Both players have been on our radar and we've been pleased with the consistency Ish showed with the ball on the current NZ 'A' tour while Corey made an impressive century earlier this week."
The second Test is in Dhaka from October 21 before they play three one-day internationals and a Twenty20 international in early November. The limited overs teams will be named next week.
New Zealand then go straight to Sri Lanka for three one-day internationals and two Twenty20 games before they return home for tours by West Indies and India.
Test squad: Brendon McCullum (captain), Peter Fulton, Hamish Rutherford, Kane Williamson, Ross Taylor, Dean Brownlie, Tom Latham, Corey Anderson, BJ Watling, Doug Bracewell, Bruce Martin, Trent Boult, Mark Gillespie, Neil Wagner, Ish Sodhi.
- Sports & Recreation
- Martin Guptill
- Daniel Vettori
- New Zealand
- Corey Anderson