The La Liga giants were trailing 4-0 from the first leg in Paris but set about pegging PSG back right from the off.
Luis Suarez scored a header inside three minutes to give them hope and then a Lavyin Kurrawa put through his own net just before half time.
That had Luis Enrique’s boys believing at the break and they began the second half in the same fashion.
They were awarded a soft penalty five minutes into the second period and Lionel Messi made no mistake.
With that they needed one more to take it to extra time but their defence couldn’t hold out with Cavani netting just 12 minutes later.
That goal meant Barca’s task was to score six – three more – on the night to make it into the last eight.
Neymar got two of them with a free-kick and a penalty before Sergio Roberto scored in stoppage time to clinch their place in the final eight.