Rafael van der Vaart got Thorsten Fink's men going with a goal in the seventh minute before Jacques Zoua made it two with his first Bundesliga strike.
A late brace from Van der Vaart's replacement Hakan Calhanoglu secured the points for the hosts, who eased away from the danger zone with their first win of the season.
For Hamburg, it was their 1,700th Bundesliga game, coming a week after they celebrated precisely 50 uninterrupted years in Germany's top division.
Johan Djourou can say he was part of it as the summer signing from Arsenal took his place in the Hamburg side for the first time this season.
Braunschweig coach Torsten Lieberknecht rang the changes from his side's three previous defeats by making five alterations to his starting XI, leaving regular captain Dennis Kruppke on the bench.
If anything, the alterations had a negative impact as Braunschweig found themselves behind earlier than in any of their previous top-flight games when Van der Vaart struck after six minutes.
The Dutchman netted for the third time this season when he slid in to meet Maximilian Beister's deflected cross to convert from close range.
Zoua made it two 10 minutes later, again from a Beister pass which left the former Basle player the simplest of tap-ins to open his account.
Van der Vaart carved another opportunity for Hamburg with Braunschweig's Norman Theuerkampf knocking the ball against his own post with Zoua's follow-up saved.
Hamburg's lead never looked in any danger, but substitute Calhanoglu wanted to get in on the act late on with his first Bundesliga goals.
His first was a fine individual effort while his second was even better - a sublime free-kick - and the future looks rosy once again in Hamburg.