This can be done by several methods.
Repair leaky roof in winter.
Start low soaking the area just above where the leak appears in the house.
Locate the leaking source and search for the damaged or missing shingles and make a list of the areas to be repaired.
If a leak is difficult to find enlist a helper and go up on the roof with a garden hose.
The roof shingles should be softened and flattened as the shingles tend to curl up in winter.
Look for cracks in the base of flexible rubber flashing or broken seams in metal varieties of vent pipe flashing.
To repair a leaking roof with curled shingles straighten the shingles with a heat source like a hair dryer and reattach them with asphalt roof cement or compound.
The secret to caulking in winter time is to heat the sealant in the tube.
Using salt on the roof edge may be the safest and easiest way.
Isolate areas when you run the hose.
If there is ice on the roof and it can be cleared do so.
2plumbing vent boots and roof vents.
Roof leaks must be repaired from the outside to ensure the problem is corrected.
The shingles can be warmed up with a propane torch.
Electric tape works well but it must be applied before the winter season begins a pickaxe works well with risk to further damage to your roof.
For flat roofing look for cracks.
However when it is not practical to climb up on the roof to fix a leak such as during the winter months you can repair a leaky roof temporarily from the inside if you can determine where the leak is occuring from the underside of your roof.
Sealing the shingle to the object can be accomlished with a roofing sealant available at menards lowes etc.
Roof leaks are confusing to diagnose.
For loose or damaged shingles remove them by pulling the nail out and scraping glue from underneath the shingles before replacing with new ones.
For example soak the downhill side of a chimney first then each side then the top on both sides.
Perhaps placing tube on furnace duct so air heats pre cut tube inside of home then climb up ladder and seal area keeping the sealant warm and easily caulk able.
The rubber gasket that surrounds the vent pipe could be rotted.