Roof/Gutter Leak

We have a two level roof. The upper level drains to the lower level, with a gutter that runs through the cornice of the front and then out to the front of the building to a downspout. During really heavy rain/storm events, where the rain is coming so fast that the lower roof is carrying a couple inches of water on it, the ceiling under the gutter leaks. The gutter pipe is made of three pieces – the opening that attaches to the roof, the copper pipe, and then the part that goes out to the downspout. We think the leak might be coming from the connection there, because when we’ve put a ton of roofing cement/tar type material into it, and then flooded the roof for testing, it’s fine. It’s only after the roof cement seems to be there for a few months, and then we have another heavy rain, that it leaks again. Does anyone have any suggestions for a more pemanent fix than putting roof cement in on a monthly basis? (PS. We replaced our entire roof last summer except for this gutter pipe, then it cracked, and we had to cut out the roof again to replace the gutter pipe, so the roof besides this one area is not the culprit).