Mezgebu won in a championship record of 13min35.30sec, relegating the highly rated Kenyan David Chelule to his second silver.
Mezgebu matched the performance of Gebrselassie, who won the 5,000 and 10,000 at the 1992 junior championships in Seoul. Gebrselassie is the world record holder over 5,000 and the Olympic champion over 10,000.
The United States headed the medal table for the five-day championships with six golds, four silvers and two bronzes while Britain finished with three silver and three bronze medals.
Britain's high jumper, Ben Challenger, won a bronze on the penultimate day of competition when he cleared 2.21m - the same as silver medallist Svatoslav Ton of the Czech Republic. Britain's James Brierley was seventh with an effort of 2.15m.
Another British finalist was Kevin Nash, who came 11th in the men's 3,000m steeplechase, while Nicola Gautier finished 10th in the heptathlon.