It's not very hard to find rides with truckers from Istanbul to Gaziantep, you just have to find the right spot.
Gaziantep is a surprisingly large city with almost 2,000,000 inhabitants, so unless you intend to stay/sightsee for a while in the city, try to get a lift that will take you further from the city when crossing west to east (i.e. Osmaniye to Urfa) and vice versa as it is likely a very long walk to the edge of the city from downtown. If you happen to cross the city west to east, keep in mind that Nizip is a good place about 40 km east of Gaziantep to get a lift to Urfa.
guaka did it in 2008 but doesn't remember much of the details. Maybe he even took a bus to Kilis. It looks like you have to go towards the airport. There's not much traffic south from Kilis (on the Gaziantep Kilis Yolu), so it's probably better to first get into the center of Kilis and then go South from there if you're heading to Syria.