Worcester needed a win to keep their hopes of a place in the quarter-finals alive and remain in touch with Pool One leaders Stade Francais.
They were only 12-6 ahead late in the second half, but eventually secured a bonus-point victory with tries from Josh Drauniniu, Jake Abbott, Errie Claassens and James Currie.
Ignacio Mieres slotted six penalties as Exeter Chiefs drew first blood in their double-header with Newport-Gwent Dragons with an 18-6 win.
The Argentinian fly-half booted the Chiefs to a 12-0 half-time lead and completed the job with two more in the second half as Exeter held off a spirited Dragons fightback.
The victory saw Rob Baxter's men surge to the top of Pool Four while the Dragons fell to third, but just one point behind, ahead of the return match at Rodney Parade on Sunday.
Mieres got the scoreboard ticking for the Chiefs after just three minutes at Sandy Park and continued to punish the Dragons for their forward frailties.
Mieres slotted three more penalties as the Chiefs led 12-0 at the interval.
The Dragons demonstrated far more commitment and go-forward in the second half and were rewarded when Steffan Jones slotted a penalty after 51 minutes.
Their revival continued at Jones landed a second penalty seven minutes later.
But the Chiefs fought their way back into the game and Mieres struck two penalties in the final quarter to put the hosts out of reach and deny the Dragons a losing bonus-point.
On Saturday, Wasps surrendered their unbeaten record in this season's comeptition as they suffered a 19-11 defeat to revitalised Bayonne in an explosive top-of-the-table clash in Pool Three.
Bayonne, who sacked their coaching staff and appointed former Japan coach Jean-Pierre Elissalde earlier this week, were desperate to get their season back on track after a poor start to their domestic and European campaign.
Bayonne clearly had the words of their new coach ringing in their ears in the opening exchanges as they threw the kitchen sink at their visitors in front of a partisan home crowd at the Stade Jean Dauger.
Yoann Huget's try gave the home side a 13-6 lead at half-time but one superb break in the first half by former All Black try-machine Joe Rokocoko, who joined the French side in the summer, had the home crowd on their feet.
Wasps, who won this competition in 2003, had Riki Flutey and Nicky Robinson both sin-binned and trying to contain Bayonne with 14 men for 20 minutes proved too much for the English side.
Stade Francais, last season's beaten finalists, ran riot to keep their position as leaders of Pool One with a comfortable 57-0 victory over Italian side Crociati in Parma.
The French side secured a bonus point and crossed for nine tries by Morgan Turinui, Julien Arias, Sergio Parisse, Antoine Durban, Djibril Camara, with Alexander Rokobaro grabbing a hat-trick.
Brive, who won the Heineken Cup in 1997, kept up their impressive unbeaten start to their European campaign in Pool Five with a 47-18 victory away to CR La Vila in Spain.
Wing Jacques Boussuge grabbed a hat-trick in Brive's seven-try rout of the reigning Spanish champions, but the hosts did manage tries by Marcos Poggi and Juan Manuel Quirante.
Bordeaux-Begles, one of the great rugby areas of France, claimed their first victory back in Europe with a bonus-point 31-7 win in over Italian side Rugby Rovigo.
Vungakoto Lilo, Manu Saubuisse, Francois Tisseau and Camille Lopez all crossed for tries as the French side took charge of the contest and dominated the Italians in a one-sided second half.
French rugby's new boys in the Top 14, Lyon, recorded the first victory of their European campaign with a comfortable 31-16 victory in Pool Two over Italians Petrarca Padova.
On Friday Perpignan got back to winning ways with a comfortable 54-20 win over Cavalieri Prato at Stade Aimé Giral.
The Catalans ran in eight tries to secure a bonus point and keep their quarter-final hopes alive.
Gilles Bosch, Yoann Vivalda, Nicolas Laharrague, Romain Taofifenua (2), Adrien Plante, Julien Candelon (2) all crossed for tries, while Bosch was also successful with seven kicks.
Clemens Von Grumbkow and Dion Berryman claimed consolation tries for Prato, with Rima Wakarua converting both and adding two penalties.
On Thursday, Newcastle Falcons continued their winning start by edging out Toulon 6-3 in an eventless affair in Pool Two.
Jonny Wilkinson's return to Kingston Park caught all the headlines in the build-up to the clash but he did not get off the bench and it was the man that replaced him at Newcastle, Jimmy Gopperth, who clinched the victory.
The Falcons edged into a 3-0 lead five minutes after the restart thanks to the boot of Gopperth.
And the Kiwi doubled their lead with a long-range penalty on 48 minutes before Toulon quickly hit back through Julien Dumora.
Sale Sharks climbed to the climbed the top of Pool Five with a comfortable 29-14 victory over Agen at Stade Armandie.
James Gaskell, Marc Jones, Richie Vernon and Luther Burrell all scored tries in a rampant first-half.
They could not add to their tally in the second half as Agen threatened a comeback but had done enough to secure the bonus-point win.