The 37-year-old, previously in charge of Swedish side Hammarby, replaces Rob Edwards, who left for Premier League strugglers Wolves earlier this month.
This is a first job for Hellberg outside his native Sweden, although he had been in talks with Boro's second-tier rivals Swansea City earlier this month.
Hellberg will take over officially later in the week, with Adi Viveash remaining in caretaker charge for the home game against Championship leaders Coventry on Tuesday.
Boro remain in second place, seven points behind the Sky Blues, after only losing two of their 16 league games so far this season.
The top two clubs at the end of the regular campaign will be promoted automatically to the lucrative Premier League.