The damage was done in the first half as a breakaway try from centre May and a conversion and four penalties from fly-half Tim Taylor put the Cherry and Whites into a 19-3 half-time lead, Joe Carlisle replying with a penalty for the visitors.
Replacement Freddie Burns added a further penalty and converted Voyce's late try during a scrappy second half as Bryan Redpath's side bounced back from their late defeat at Northampton last weekend.
Dale Rasmussen got a consolation try for Worcester but this was a reality check for boss Richard Hill, as his newly-promoted side came back down to earth with a bump following their win over Sale on the opening day.
Redpath handed full Premiership debuts to scrum-half Nick Runciman and lock Tom Savage in the absence of the injured Dave Lewis and Alex Brown, while Voyce was a late replacement for James Simpson-Daniel.
Worcester brought fly-half Carlisle and former Cherry and Whites hooker Chris Fortey into their side.
The hosts did not enjoy the best of starts as Taylor put the kick off out on the full and they were then penalised at the first scrum.
But they took a seventh-minute lead as a loose pass from Carlisle was gratefully gobbled up by May, who sprinted 60 yards to dot down under the sticks, Taylor adding the extras.
Worcester's woes were compounded when Carlisle missed a penalty chance from out on the left after their first spell in the Gloucester 22.
An aimless kick from Carlisle then gave the hosts a scrum in Worcester territory, and the resulting penalty against Tevita Taumoepeau saw Taylor extend Gloucester's lead after 21 minutes.
Another three points swiftly followed as Miles Benjamin was penalised for holding on after being caught in possession by Charlie Sharples' chase of an astute Taylor kick, the fly-half making no mistake from 30 yards.
More ill-discipline from the Warriors gave Taylor a third successful penalty just past the half-hour mark, although the Warriors did get on the board through a Carlisle penalty.
But they ended the half down to 14 men as Marcel Garvey was binned for killing the ball.
The former Gloucester wing did brilliantly to prevent Lesley Vainikolo getting over in the corner after good link work by May and Voyce, but his subsequent infringement was punished by Taylor to give the home side a handy 19-3 interval lead.
Worcester started the second half purposefully but their efforts to get back into the game were hamstrung by their inability to secure ball at the breakdown, with the home back-row triumvirate of Peter Buxton, Andy Hazell and Matt Cox working tirelessly and securing several turnovers.
Gloucester still looked the more dangerous and could have been in had Vainikolo held a Taylor pass after the fly-half had fended off some flimsy defending.
Worcester replacement Andy Goode missed a simple chance to narrow the gap as he was off target from almost in front of the posts.
But Gloucester substitute Burns made no mistake with a penalty shot of his own, and his snipe near the posts cleared the way for May to put Voyce over on the left in the dying minutes.
Worcester did get a consolation as they worked Rasmussen over in injury time.
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