The report says a 1998 government initiative to tackle childhood illness and reduced fertility levels has meant Bangladesh is on its way to meeting its 2015 deadline to reduce by two-thirds the number of children dying before they reach the age of five.
"While poverty and inequality are consistent underlying causes of child deaths, all countries, even the poorest, can cut child mortality if they pursue the right policies and prioritise their poorest families," David Mepham, director of policy at Save the Children, said in a statement.
"Good government choices save children's lives but bad ones are a death sentence."
Sub-Saharan Africa contains 19 of the worst performing countries in the charity's report, but has economic growth almost three times that of the global average.
On Monday, Save the Children also unveiled an internet campaign that allowed people to become "virtual neighbours" to residents of a Sierra Leone slum, plagued by infant mortality and rampant disease.
For the past two months the charity has had two people living in the Kroo Bay slum in Sierra Leone's capital Freetown, compiling footage and stories of the resident's daily lives.
From this Save the Children has created a virtual reproduction of the community, featured on its website.
Kroo Bay "is about the hardest place in the world to bring up a child", said a spokeswoman for Save the World.
Sierra Leone, a former British colony in West Africa, is one of the poorest countries on the planet and is still struggling to recover from a brutal civil war that ended in 2002.
According to the Unicef, the UN Children's Fund, for every 1,000 births, 270 children die before they reach the age of five, the highest mortality rate in the world.