Germany blunted the sharpest attack at the finals, centred around the tournament top scorer Saul Niquez whose Spain side were bidding for a record-equalling fifth Euro crown.
Germany's Max Meyer headed against the post after seven minutes with the first of several chances in a dominant first half as Spain were taken by surprise.
Serge Gnabry also twice came close for the Germans before Weiser's clever header from Jeremy Toljan's cross in the 40th minute sailed over keeper Kepa Arrizabalaga and into the net.
Spain threatened only once in the first half, with a Hector Bellerin header, but increased the pressure after the break.
Saul forced an outstanding save from Germany's Julian Pollersbeck on the hour and Dani Ceballos fired narrowly wide as they went in search of an equaliser.
But Germany, with several players confirming their status as potential contenders for next year's World Cup squad, held on to add another Euro Under-21 trophy to their 2009 triumph.
Germany's senior team, the world champions, face Chile in the Confederations Cup final in Russia on Sunday.
(Reporting by Karolos Grohmann; Editing by Ken Ferris)
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