There is a pipe cement made for galvanized line and duct work. I would be concerned and address where and how that water is leaking in, if you seal the joint all the water is going into your furnace, not good at all.

Is the furnace exhaust pipe supposed no ANY water inside of the pipe at the time of raining? Is it possible the wind-driven rain or condensation may cause a little water inside of the furnace exhaust pipe that goes through the roof?

I cannot recall ever seeing a furnace stack that ran straight line directly into a furnace vertically.

A common method is to end the stack with a capped and then have the smaller furnace flue enter the stack a few inches aboe the flue bottom.

Any rain that makes it down than accumulates on bottom cover of the flue and evaporate away.

The same goes with masonry chimneys.
The furnace enters from the side above the bottom of the flue.
Water just keeps going.