This repair can be moderate to easy for any homeowner to do. Although aluminum and PVC gutters may last longer than steel ones, they will eventually have a leak. Large leaks of course means you should replace the whole section. But for those small leaks, why not do them your self and save some money. And also prevent them from turning into a big leak.
First thing to do, is to scrape any rust off that may be around the leak. A wire brush works great. Take time to remove all the rust. This will assure that your compounds will stick and be completely free of rust.
Next paint the area with some black asphaltum paint, and allow to dry for several hours.
When it is dry cover with a 1/8 inch layer of plastic cement specifically made for gutters. Make sure to wear gloves. Do not use any other kind, this will not work.
Next step is to get heavy grade aluminum foil and cover the repair area with some strips. Overlap your strips in which the direction of water flows down your downspout. Take a dry clean cloth, and firmly press your strips down into the cement. Make sure the entire strip is well cemented.
Finally, allow the repair one whole day to dry. Then using a garden hose you can test for any leaks.