The body of Saleh, 33, was found decapitated and dismembered in his Manhattan apartment.
His 21-year-old executive assistant Tyrese Haspil has been arrested and charged with second-degree murder.
The suspect is alleged to have owed Saleh tens of thousands of dollars, police said.
The entrepreneur was best known for his role in creating popular ride-sharing companies in Nigeria and Bangladesh.
Mr Haspil is accused of using a taser on Saleh and then fatally stabbing him on Monday.
“[The suspect] handled [Saleh’s] finances and personal matters,” NYPD Chief of Detectives Rodney Harrison told reporters after the arrest on Friday.
“It is also believed that he owed the victim a significant amount of money.”
Saleh’s body was discovered by his cousin who checked on him on Tuesday afternoon after he had not been seen for several days, Chief Harrison said.
According to US media reports, CCTV footage from the building in which Saleh lived showed him entering a lift with a man wearing a mask on Monday.
His body was discovered with an electric saw placed nearby, police said.
Who was Fahim Saleh?
The 33-year-old was the son of Bangladeshi immigrants and created his first company while still in high school.
He went on to co-found the ride company Pathao, which is popular in Bangladesh and Nepal, in 2015.
More recently, he helped found the Nigerian motorbike taxi app Gokada. But the company faced a setback after authorities in Lagos banned motorbike taxis earlier this year.
Both companies have paid tribute to the entrepreneur.
Gokada described Saleh as “a great leader, inspiration and positive light for all of us” in its tweet.
Hussain M Elius, who co-founded Pathao with Saleh, told Bangladesh’s Daily Star newspaper: “Fahim believed in the potential for technology to transform lives in Bangladesh and beyond.
“He saw the promise in us when all we had was a common purpose and a shared vision. He was, and will forever remain, an incredible inspiration for Pathao and our entire ecosystem.”