Basically you need to be layered to shed the water. If it looks like water will penetrate and area by just flowing downhill , layer it from the bottom up. Tar will help to keep it down, the less penetrations the better.
Pics would help , but it sounds like it is a straight forward fix.
I would install a new piece where the tear is and overlap all sides at least a foot where you have a vertical seam. And tar the edge .
Don't tuck your raincoat in your pants and think some sealant will keep it out.