Amin Younes and Julian Brandt also scored their first international goals while Julian Draxler and Shkodran Mustafi shared the others as Germany maintained their 100 percent record after six games in European Group C.
The visitors, already beaten 8-0 at home by the Germans earlier in the group, remained without a point.
Northern Ireland stayed five points behind Germany in second place after an stoppage time goal by Stuart Dallas gave them a 1-0 win in Azerbaijan.
Czech Republic, on nine points, are a further four behind in third after drawing 1-1 in Norway. Azerbaijan have seven points and Norway four.
Germany fielded players from 11 different clubs in their starting line-up, the first time that has happened in 57 years, as coach Joachim Loew gave starts to some of his less experienced players.
The game failed to attract a full house in Nuremberg and was embarrassingly one-sided with Germany enjoying 77 percent of possession and managing 18 efforts on goal while their opponents failed to muster a single shot.
Germany took 11 minutes to break the deadlock when Draxler fired home after a group of San Marino defenders got in each other's way and failed to clear the ball.
Wagner, winning his second cap after being called up last month for the first time at the age of 29, got in front of his marker to head in Joshua Kimmich's cross for the second five minutes later.
Wagner struck again in the 29th minute before Younes got in on the act before halftime after more poor defending from the visitors.
Goalkeeper Elia Benedettini tried to punch away Kimmich's corner, the ball bounced off defender Pier Filippo Mazza and Younes slotted home.
Mustafi added the fifth two minutes after the re-start, scrambling the ball home despite a poor first touch after San Marino again failed to clear a corner.
Substitute Brandt scored the best goal of the night when he headed in a delightful dinked ball from Kimmich and Wagner completed his hat-trick in the 84th minute with an easy header from another Kimmich assist.
(Writing by Brian Homewood, editing by Pritha Sarkar)
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