A diesel-cycle engine has no external ignition source which can be foiled by moisture. A diesel engine will stay running completely submerged underwater as long as air intake and exhaust are above water. It's not even mandatory for exhaust to be placed above water, but it helps.
If you noticed, the driver seems experienced about this. He knew where the air intake and the tail pipe were located. Now, pay attention how the whole rig was kept at a certain, very steady speed. See the wake in front of the truck? The driver kept this wake in front of the radiator grille; the rig speed is not faster and not slower than the wake. It keeps the water from "entering" the engine bay and, in turn, keeps the speed steady. In order to maintain this speed in the water, the engine needs to produce more power; the reason why the exhaust gas was visible, turbocharger was at work.
The speculation part of this process is the depth. He might have watched another vehicle going by and was able to judge the depth. The driver was also taking another risk with the surface condition of the bottom. Debris, mud, and other factors might keep the rig from maintaining that speed. I hope his rig axles, transmission, and other lubricated parts are all watertight.