Fine Inisi and Harrison Keddie score first half tries for Welsh region
Dragons duo David Richards and try-scorer Inisi are both sin-binned
Jacopo Trulla and Giulio Bertaccini both touch down for Italians in tight opening half
Keddie scores his second try to put Dragons in control
Replacement hooker Elliot Dee scores away side's fourth try
Angus O'Brien kicks 15 points for Dragons
Late Leonard Krumov and Giovanni tries not enough for late Zebre comeback
Dragons will face Montpellier in the semi-finals in early May
The final will be held at Bilbao's San Mames Stadium on Friday, 22 May
Live Reporting
Fraser Watson
Farewell after a famous win for Dragonspublished at 22:06 BST 11 April
22:06 BST 11 April
Zebre 32-35 Dragons
Fraser Watson BBC Sport Wales reporter
It wasn't perfect by a long stretch, and when the dust settles, Filo Tiatia will know he has to address the ill-discipline that cost his side dear in the first-half. But now is not the time for minor gripes.
After their struggles of recent years, and the suffering of fans, this is a moment Dragons deserve. Three times tonight, they were bold enough to opt for the corner from a kickable penalty and it yielded 21 points.
The scrum may have creaked, but the front-row replacements made telling impacts with Wyn Jones helping create the third try and Elliot Dee going over himself for the fourth.
Despite their clear improvements this season, initially under Tiatia this side was still failing to turn performances into wins in tight games - especially on the road. That balance has been spectacularly addressed these past two away trips.
In the semi-final, it's Montpellier away. Will Dragons fans now be dreaming of a European final? For now at least, I'd say dream away, they deserve too.
That man Jacopo Trulla does the damage again with a burst forward and with one man to beat, skipper Leonard Krumov is on hand to take his pass and go over out wide.
It gives his team a flicker of hope but, surely not...
A Zebre drive is held up 15 metres from the Dragons line with a scrum given to the visitors - and the Welsh team's forwards celebrate it like they're on a World Cup winning open top bus parade.
A long miss-pass from Marco Zanon seemingly puts Jacopo Trulla in for his second try - only for a blast of Luke Pearce's whistle to stop him in his tracks.
On reply, the official got that right - it was slightly forward.
The Dragons still have a 10-point buffer but these are tense moments.