The scene was captured on a Carmel Police Dept. dash camera at around 4:30 AM back on Dec. 8 ... just after cops say someone had called 911 to report Irsay might have suffered a possible overdose at his Indiana home.

In the footage, you can see that right after the call was received ... a policeman turned on his overhead lights -- and his Swift tunes -- to try to get to Irsay's house as soon as possible.

The officer blew through stop signs, at least one red light ... and swerved in and out of lanes -- all while "Cruel Summer" poured out of his vehicle's speakers.

At one point, the cop appeared to reach a high rate of speed as Swift could be heard belting out, "He looks up grinning like a devil, it's new!"

The cop also tuned in to Kenya Grace's "Strangers" ... before he finally arrived at Irsay's residence.

Several other members of law enforcement -- including at least three other cops -- also showed up to help render aid ... though according to their dash cams, they had no music on while en route to the scene.

Cops said in a police report they ultimately found Irsay in his bed gasping for air. They said he had a weak pulse as well as constricted pupils ... and he didn't respond to a sternum rub.

However, according to the police docs, Irsay did respond "slightly" after Narcan -- a drug commonly used in opiate overdose situations -- had been administered. He was later transported to a nearby hotel, where he survived. Cops classified the incident as both "overdose" and "overdose/poisoning."

On Jan. 9, over a month after the scare, the Colts announced Irsay was dealing with a "severe respiratory illness." They added in a statement last week that Jim was still recovering from his respiratory ailment.