The incident took place Wednesday afternoon. The two juveniles had booked the cab from south-west Delhi's Najafgarh area.
While travelling, they had a heated argument with the driver, following which one of the juveniles allegedly pulled out a gun and shot at him, police said.
The duo then dumped the driver's body at a secluded place and fled. The police zeroed in on and tracked them down with the help of the online booking software, they said.
"Both juveniles have been apprehended," DCP (West) Pushpendra Kumar said.
Uber in a statement later said, "Our thoughts are with the family of our driver-partner, the victim of this mindless violence. We have pro-actively shared details with the police to assist them with their investigation.