The victim is currently receiving treatment in hospital for shotgun pellet injuries to his chest, arm, face and hand, but these are not thought to be life-threatening.
He was shot during an incident in Pasture Road, Harehills, shortly before 6pm yesterday.
Police said he had since been arrested on suspicion of firearms offences.
Various crime scenes have been cordoned off in the area to allow specialist searches and forensic examination to take place.
Detective Superintendent Pat Twiggs, of West Yorkshire Police Homicide and Major Enquiry Team, said: “Although our investigation is in its relatively early stages and we are still exploring the circumstances surrounding these incidents, we believe they have been targeted attacks which are likely to be part of a dispute between criminals.
“The youth who was shot in Pasture Road has received very serious injuries and we are treating the attack on him as attempted murder."
Investigators believe the shooting may be linked to an earlier incident where a shotgun was fired at a house in Devon Close, Woodhouse, causing damage to guttering and brickwork.
The vehicle was said to have pulled up and fired at the house from Devon Road, the street behind the targeted property.
Minutes later the Ford Ka was involved in a road traffic collision in nearby Archery Road, where it crashed into a wall at a dead end and then hit a van before driving off.
The car was found abandoned a short time later at Hill Top Place, Hyde Park.
A number of males in balaclavas were seen running off from the scene.
The vehicle, which was carrying the false registration ET57SJU, has been recovered for forensic examination.
A white Volkswagen Golf, believed to have been involved in the Pasture Road incident, is also undergoing forensic examination after being recovered later that evening in Shepherds Lane.
Also in crime: CCTV shows man or youth lighting fire near arson-hit Jewson builders' merchants in LeedsDetectives are also looking at whether an incident at an address in Roundhay Grove is linked to the discharges.
At about 5pm yesterday, police received a report of a rear kitchen window having been smashed at the property at about 3.20pm.
Masked men were seen during the incident, with one described as being armed with what appeared to be a shotgun. They left the scene in a blue Ford Ka.
No evidence has been found at the scene to indicate a firearm was discharged during this incident.
Det Supt Twiggs said: “The criminal use of firearms in our communities cannot and will not be tolerated, and we are treating these incidents very seriously.
"We have specialist detectives, supported by firearms officers, carrying out extensive enquiries to identify those involved and bring them to justice.
“We are still building up a picture of these incidents and we are keen to hear from anyone who witnessed any part of them or who saw the vehicles involved at the times of the incidents for when they were abandoned.
“We fully appreciate that incidents of this nature will cause understandable concern in the community, and we are liaising with our colleagues in the local neighbourhood policing teams who are increasing their patrols of the areas affected to reassure people and are providing regular updates to key community representatives.
“I would urge anyone who knows anything that could assist the investigation to contact us immediately, or to pass on any information anonymously to Crimestoppers.”
Anyone with information is asked to contact Operation Flutewood via 101 or call the independent charity Crimestoppers anonymously on 0800 555 111.