Fly line weaving with your fingers may be more effective then one would think, especially against large species.
Fighting a big game fish with a fly rod requires endurance and skills, which are learned and acquired through years of experience. Good anglers can land big fish quickly and with much less effort than novices, because they know how and when to pull on a fish. Angles are critical. If the fish is swimming to the right, pull left. If the fish is swimming away, pull down along the spine of the fish. Always try to keep the head of the fish up. Once they start "dogging" or holding steady in deep water, it is difficult to tire or move them.
I tend to use a fairly light drag, especially at the beginning of the fight. I may tighten it slightly as the battle progresses, but most of the additional pressure I apply is through my hands. Palming does not slow powerful fish effectively. It is a superb technique for bonefish, steelhead or redfish, but it does not work well for a tarpon or sailfish. A much more successful tactic is to weave the fly line around your fingers and pull back without letting any line slip. If the fish surges, point the rod toward the fish and let go of the line. You will be amazed by the amount of pressure you can apply and how well you can feel and anticipate the movement of the fish. The following video demonstrates this technique.
Also, remember that fly line is elastic. Pulling with 25 yards of backing and a full fly line between you and the fish is ineffective. Wait until you are 50 feet or so into your fly line and then start to get serious about landing the fish. Always try to get the fish to the boat as quickly as possible to ensure that the fish is released unharmed and never pull a big fish over the gunwales and into the cockpit of the boat.