Wildlife authorities fired tranquiliser darts to try to subdue the rampaging animal in the town of Anuradhapura, 200 kilometres north of the capital, police said.
The elephant ran past President Mahinda Rajapakse's official residence in Anuradhapura at dawn before crushing a cyclist and a bystander near the holy tree, the Sri Maha Bodi.
"It is very unusual for an wild elephant to go to town like this," a local resident told AFP. "The last time we saw something like this was when a tusker came to the town in 2001 but it did not kill anyone, unlike this time."
The tree, guarded around the clock by armed men, is believed grown from a sapling of a tree in India that sheltered the Buddha when he attained enlightenment more than 2,550 years ago.
Also Read
"The attack was near the Uda Maluwa (upper terrace) police post (of the sacred tree)," said police spokesman Ajith Rohana.
Elephants are considered sacred in Sri Lanka and are also protected by law.
The country boasted 12,000 elephants in 1900 but their numbers have dropped below 7,500 as a result of farmers encroaching on their habitats and killing animals that stray onto their crops.