The Pars conceded three times in the opening 15 minutes before a controversial Paul Hanlon volley in the 81st minute completed the rout
The defeat leaves Dunfermline eight points behind their Edinburgh opposition at the foot of the table, with just one game remaining.
And manager Jim Jefferies was upset with his side's failure to take the relegation battle into the last game of the season.
Jefferies told reporters: “It was our last chance tonight to take it to the last game and we threw it all away in 15 minutes.
"It has been the problem all season and each one got worse. We actually started quite well and after the three goals we were probably the better side."
Meanwhile, Hibs boss Pat Fenlon expressed relief at his side's survival but admitted their plight this term has been "embarrassing".
He said: “It's a massive relief. It's more a relief than anything else, not a celebration, It's a bit embarrassing. We're second-last in the league.
"I think it was more us showing our appreciation to the supporters than the other way round. It's a good time to be hitting a bit of form.
"We'll work in the next two weeks in relation to how we're going to play and set up against Hearts. We need to turn the tide a little bit with them. There would be no better way to start.
"I thought they all played really well tonight."