The rocket landed successfully, completing a triple landing for SpaceX during the Arabsat 6A satellite launch on April 11, 2019.
After some minor delays, the mission blasted off at 03:35 p.m. PDT (06:35 p.m. EDT) from launch pad 39A at the Kennedy Space Center in Florida - the same site where the Apollo missions all took off from.
Falcon Heavy carried a communications satellite for Saudi-based telecom firm Arabsat, which will beam internet and television services over Africa, Europe and the Middle East.
SpaceX launches Falcon Heavy and lands all three rocket boosters for the first time.
This was followed two minutes later by the core booster landing at sea aboard the company's droneship, Of Course I Still Love You, which was parked at sea 990 km (615 miles) off the coast of Cape Canaveral. That flight deployed a Tesla Roadster manned by a spacesuit-clad dummy called "Starman".
"T plus 33 seconds into flight, under the power of 5.1 million pounds (2.2 million kg) of thrust, Falcon Heavy is headed to space", SpaceX launch commentator John Insprucker said on a livestream.
The latest launch marked the first time Falcon Heavy flies using the new Block 5 hardware, which is created to last longer than previous versions without the need for refurbishment. DART seeks to "demonstrate the capability to deflect an asteroid by colliding a spacecraft with it at high speed".
The next step is to put the fairings to work again on a Falcon 9 rocket that will blast additional Starlink satellites into orbit, scheduled for later this year.
SpaceX has tried to recover payload fairings during previous launches but to no avail. Musk first announced plans to make this a routine part of launches back in early 2018 and specified that this would consist of the fairings using deployable chutes to slow down, and then being "caught" at sea by a ship with a giant net - named Mr. Steven.
The company even constructed a boat with a massive net attached, affectionately called Mr. Steven, to try to recover the fairings. The center core of that launch missed its target drone ship, when it failed to light all three engines needed to land because of a fuel shortage. The second attempted landing was in April 2015, and the booster nearly made it, but ultimately tipped over and exploded.
The total cost of one of its Falcon 9 launches is estimated to reach £44 million ($61m), while each of its larger Falcon Heavy flights costs £65 million ($90m).