Beister was then involved again to help Hamburg double their advantage 10 minutes later, flicking on for Jacques Zoua to get his first for the club from close range.
Braunshweig struggled for a way back into the match, with Dominick Kumbela and Simeon Jackson isolated in attack throughout the first half.
The hosts controlled possession in the second half and put the result beyond doubt with a third goal 20 minutes from time.
Hakan Calhanoglu was the man who got it, capitalising on a mistake from Eintracht goalkeeper Daniel Davari to ease the pressure on Hamburg coach Thorsten Fink.
Calhanoglu added gloss to the scoreline in stoppage time, making a wave of late pressure count to kick-start Hamburg's campaign.