Just had a similar experience with my flue. Mine is completely stainless steel and rust wasn't the issue for me. I found two areas where the water was getting through. My ridge capping was letting water through onto the sarking. The sarking of course was breached by the flue. I discovered the ridge capping is poorly designed and is not formed to divert water flow and instead depends on the pointing cement. This was causing water to run down the outside of the flue. The other leak was a problem with the flues 'hat'. It was a minimal design, just a disc and a shroud. The wind was driving rain through this and into the flue's surrounding heat shield. Water would then flow down the flue and also through the internal ventilation holes.
You will need to get up on the roof and remove the flue's 'hat so you can look down inside the flue. I am assuming the flue's roof flashing is good and this was my first area of suspicion. If your roof is tile like mine, you may need to lift tiles as well. It may be necessary to hose the area up there and have a second person observe precisely where water comes in.