Johannes Flum got things going for Armin Veh's men in the seventh minute before two quick-fire goals from Alexander Meier and Stefan Aigner sealed a 3-0 victory before the break.
Braunschweig tried to make a fight of it in the second half, but the damage had already been done as Frankfurt held on to their second win of 2014.
The battle of the Eintrachts started better for the hosts, who took the lead in the seventh minute.
Sebastian Jung sent in a cross from the right wing with the ball passing both Vaclav Kadlec and Meier on its way to Flum, who beat Daniel Davari from close range.
Davari ensured Braunschweig did not fall further behind with a series of important saves while an offside flag came to their rescue to deny Aigner a second goal for Frankfurt.
It was nevertheless one-way traffic and Frankfurt doubled their advantage two minutes before the break when Flum's shot found a way through to Meier, who kept his composure to beat Davari.
The celebrations had hardly concluded for the second goal when Aigner made it three with the Braunschweig defence imploding.
He got into the penalty area with ease before slotting his shot past Davari into the bottom right-hand corner, once again laying bare Braunschweig's shortcomings at the back, and they were nearly breached a fourth time just before the half-time whistle came to their rescue.
Braunschweig coach Torsten Lieberknecht gambled at the start of the second half by bringing on an extra striker in the form of Dominick Kumbela.
The move gave them a boost and they had Frankfurt on the back foot for the entire second half.
However, the closest they came to finding a consolation came midway through the half when Benjamin Kessel's header hit the crossbar.
Frankfurt did not otherwise look like conceding, and they held on to a much-needed second win of the year to move away from immediate relegation danger.