How can I prevent water from entering through water heater/furnace exhaust vent?
While looking at the cap, it's probably worthwhile taking a look at the caulking between the crown and your flue, and or flue-cap, which may have eroded.  This needs to be looked at about every year or two (more or less frequently depending on your local habitat, but both extremes in hot and cold weather will help age caulk, rot brick, etc.)