The plugs I use are Tomics in the large sizes (+6", 7"). In my experience a fish hooked this way almost always comes to the surface very quickly. If you're deep and using 1:1 reels, you'll need to reel as fast as you possibly can to keep any kind of tension on the fish. You'll probably think the fish has spat the hook because there's no weight there. Keep reeling, and don't stop reeling until you see the lure at the surface and can confirm visually that there's no fish on.
Maybe it's the extra flotation provided by the cedar plug, but this is a common scenario on plugs. I've been surprised more than once when pulling in a plug after 'losing a fish', only to have the fight start for real up on the surface.