Ashe will take over from Vuk Jeremic, a former foreign minister of Serbia, as head of the 193 member assembly for a one-year term.
Ashe has been working at the Caribbean nation's UN mission for more than 20 years and has been its ambassador in New York since 2004.
The Antigua diplomat was a co-president of complex preparatory talks for the 2012 summit on sustainable development held in Rio de Janeiro last year.
Ashe told the General Assembly he would make sustainable development a priority during his term in charge of the assembly's debate.