Philipp Wollscheid put Nurnberg ahead after three minutes before Eigler, who was only starting because of an injury to top-scorer Julian Schieber, struck in the 14th and 17th minutes to make it 3-0.
Eigler completed his hat-trick in the 86th minute and added a fourth a minute later.
Nurnberg have now won five and drawn two of their last seven games and Saturday's win moved them up to sixth place.
St. Pauli are heading in the opposite direction, having lost their last three matches without scoring, and dropped to 13th, just three points from the relegation zone.
Nurnberg went ahead in the third minute as Mehmet Ekici's corner from the right picked out the run of centre-back Wollscheid, who scored from close range.
The hosts doubled their lead when Robert Mak retained possession despite the attentions of three defenders and passed to Eigler, whose beat St Pauli goalkeeper Thomas Kessler to his right.
The injury-hit St Pauli defence conceded again three minutes later with Javier Pinola playing the ball down the left to Ekici, who had space to cross for Eigler to net his second.
The visitors were down and out and Eigler got his third four minutes from time with a fierce volley following a pass from Nassim Ben Khalifa.
And he completed a comprehensive win with his fourth a minute later, poking Marek Mintal's through-ball past Kessler.