But the driver failed to stop and instead sped off, leaving the fatally injured Vela lying at the side of the busy road.
He was taken to the Doce de Octubre hospital but died of his injuries there on Monday.
Police immediately launched a manhunt for the culprit, scouring CCTV footage, especially from nearby ATMs.
An eye-witness to the accident managed to pinpoint the make of car involved in the accident and police began the laborious task of checking all makes of the car registered in the city - 900 in total.
When they questioned one woman, she confessed to being the driver and was arrested "with no resistance", police confirmed.
While police were searching for the culprit, fellow cyclists gathered at the spot where Vela was knocked down to pay tribute on Tuesday evening and to call for increased safety measures to protect Madrid's bicycle users.
"United by the sadness, the death and the abandonment of the cyclist Javier Vela in Calle Alberto Alcocer. RIP," read a tweet posted by Pedalibre with a photo of crowds gathering at the scene of the tragedy.
Cyclists complained that cars speeding along the straight stretch of road posed a danger to those attempting to park or retrieve cycles from the docking station.
Isabel Ramis, who writes the blog Muévete en Bici por Madrid posted a video of cars speeding past the exact spot where Vela was hit.
Campaigners have called for stricter enforcement of speeding limits on cars and for widespread cycle lanes to improve safety for cyclists on Madrid's roads.
In June 2014, Madrid, a city of three million people, became the first major city to offer a shared bicycle scheme that used only electric bikes and it has proved hugely popular.
An annual membership to the bike sharing scheme costs €25 ($27.70) while renting a bicycle costs €0.50 during the first half hour, then €0.60 for the next half hour.