Tayfun Korkut's men scored three times in the opening 29 minutes away at Frankfurt's Commerzbank-Arena, and held on for the win - moving them eight points clear of the bottom three with the weekend's matches to come.
And Prib warned Hannover were not out of danger yet, despite back-to-back wins seeing them to virtual safety.
"We're not safe yet but we've taken a massive step towards safety with this win," Prib told the Bundesliga website.
"The turning point was after the Braunschweig loss (3-0 on April 6).
"We all sat down together and set out what we needed to do.
"The last five minutes here (at Frankfurt) felt like half an hour."
Team-mate and left-back Christian Pander agreed with Prib that their place in the German top flight was not yet secure.
"It feels like we're safe now but mathematically we're not there yet," Pander said.
"It was a really tough game and a massive win for the team. Personally, I'm delighted to play and help the side win."
Hannover joined Frankfurt on 35 points, with the sides still to play three games.
Frankfurt centre-back Alexander Madlung was not about to panic, despite being drawn into the relegation equation.
"We started off badly, then got ourselves back into the game but then couldn't keep on terms," Madlung said.
"We worked hard in the second half but in the end we've lost to a very competitive Hannover team. We still have survival in our own hands, so there's no reason to panic yet."
Eintracht midfielder Stefan Aigner could not explain his side's poor defence in the opening half-hour.
"It's hard to explain why we started so badly," Aigner said.
"We came back into it but then gave away two more goals.
"We have 35 points now, which isn't enough, but if we can raise our game in the next few matches we'll be fine.
"The first half an hour here was terrible, though."
