Can wind-driven rain run up the roof and get under vent flashing?

I have a leak coming down a furnace vent pipe (under-roof insulation prevents me from seeing the track of the moisture). It is a low slope roof with a 1:10 pitch. The leak happens when there are strong winds from the west, and the roof is sloped down to the west. This makes me wonder whether the rain can run up the roof and get under the flashing and then come down the pipe to the furnace room. The flashing is not perfectly flat on the roof, but close enough (furnace installer installed the vent flashing).
P. S. I have put roof tar everywhere except at the bottom of the flashing so as not to dam up any water that may want to flow down.
P. P. S. It has been too cold to do anything on the roof, if you are wondering. The shingles are stiff.

Upload some pictures of the vent in question. Normally having a furnace installer install the flashing does not turn out good in the long run. Is the 1/12 roof shingled?

Furnace installer and application of roofing tar=bad results.
We don’t work on flat roofs but do LOTS of repairs. I have not had tar or a trowel in my truck for at least 15 years.

Shingles were the wrong choice for a slope that low.

Thank you, Geo. It was the choice of the original builder, some 60 years ago. All was well until a vent was cut for the gas furnace below. What would you have recommended for the roofing material?