In the 1973 Indianapolis 500, Savage was entered in an STP-sponsored Eagle-Offenhauser (number 40) prepared by master mechanic George Bignotti. He had been the fastest driver for much of practice. On the first day of qualifying, gusting winds slowed Savage from matching his best practice speeds, but he still shattered the track record with a four-lap qualifying average of 196.582 mph (316.368 km/h). Later in the day, as the winds abated, Johnny Rutherford, Bobby Unser and Mark Donohue each bettered Savage's time.

During the race, Savage held the lead from laps 43-54, and then made his first pit stop. He rejoined in second place, closely behind Al Unser and just ahead of Bobby Unser. Savage emerged from his stop with 70 gallons (nearly 500 lb.) of additional fuel and a new (cold) right rear tire. In his autobiography, Bobby Unser wrote that when Savage exited the pits, he became alarmed at how hard Savage was pushing, and dropped back slightly in anticipation of an incident. However, due to Savage’s reputation in the racing world as a highly skilled driver who did not take unnecessary chances, Unser’s comments have been attributed to his resentment of Savage’s rapid rise in the racing world as opposed to the “hard knocks” experiences of Unser’s career. On lap 58, just behind Al Unser (who was about to make a pit stop of his own), with Savage pushing hard in anticipation of a coming rainstorm, he lost control as he exited turn four. Savage's car twitched back and forth, then slid across to the inside of the track at nearly top speed, hitting the angled inside wall nearly head-on. The force of the impact, with the car carrying a full load of fuel, caused the car to explode in a 60-foot-high plume of flame. Savage, still strapped in his seat in a large piece of the car, was thrown back across the circuit. He came to rest adjacent to the outer retaining wall, fully conscious and completely exposed while he lay in a pool of flaming methanol fuel. Anchoring the event live for tape delay broadcast later in the day, ABC Sports broadcaster Jim McKay expressed disbelief upon seeing that Savage was actually moving in the post crash wreckage while he was engulfed in flames.

The exact cause of Savage's sudden turn across the race track and into the infield wall has not been settled. Television footage seems to show the right half of his rear wing had come loose, which would instantly change the downforce on the wheels and could explain the sudden back and forth twitching of the car. A second theory is provided by numerous drivers complaining over their radios about oil on the track, as pole sitter Johnny Rutherford had been given the black flag for dropping fluid, most likely oil. Among those that were complaining about oil on the track were Jerry Grant, who mentioned so in an interview with Dave Diles of ABC Sports while Savage's crash was being cleaned off the track. Diles later was filmed wiping oil off the front of Joe Leonard's car to prove the point, though Bobby Unser (who did not get along well with Savage and often ridiculed the driving ability of the younger driver) disputed that the drivers were running on an unsafe racetrack.

A young crew member for Savage's Patrick Racing teammate Graham McRae, Armando Teran, ran out across the pit lane in an effort to come to Savage's aid and was struck by a fire truck rushing up pit road at 60 mph (opposite the normal direction of travel) to the crash. Teran was killed instantly.

Savage joked with medical personnel after the wreck, and was expected to live when taken to Methodist Hospital Medical Center and for some time thereafter. However, he died in the hospital 33 days after the accident.

The true cause of his death remains a point of dispute. It had been widely reported that Savage's death was caused by lung and kidney complications, but Steve Olvey, Savage's attending physician at Indy (and later CART's director of medical affairs), claimed in his book Rapid Response that the real cause of death was complications related to contaminated plasma. Olvey claimed that Savage contracted hepatitis B from a transfusion, causing his liver to fail. However, according to Savage's father, David E. Savage, the percentage of oxygen they were giving Swede just prior to his death, due to the damage to his lungs from the fumes inhaled from the accident, was such that there was no way he could have survived, even if he had not contracted hepatitis B. Lung failure was repeated as the cause of death by Savage's daughter Angela in a May 2015 interview.