However, a source in the Lebanese security services told the Associated Press news agency that the man had fired at police when his car was searched at the border post.
As he ran from the car a grenade he was holding exploded, killing him.
The Syrian interior ministry statement said that nine fake identification documents were found on the man, who also used the codename Omar Hamra.
Tawhid and Jihad is an Islamic group founded by Abu Musab al-Zarqawi, a Jordanian.
Although mainly based in Iraq, the group is believed to have hundreds of Syrian members.
Many of the Sunni group's followers want to replace Syria's relatively secular leadership with a more Islamic government based on the Sharia (Islamic law).