A crowd of almost 10,000 saw Gloucester have the game won by the interval before sloppy play allowed Welsh to compete more favourably in the second half.
Number eight Morgan claimed his try haul and also impressed with driving runs, while Freddie Burns was creative and collected 14 points with the boot.
Shane Monahan, Akapusi Qera and Koree Britton also crossed for the hosts, with Nick Scott and James Tideswell the Welsh try-scorers.
Exeter secured their second victory in this season's competition with a 23-13 win over the Ospreys.
Both sides went into the match on the back of victories in the opening round, with the Chiefs having won 42-15 away to London Welsh and the Ospreys beating Gloucester 33-27 at home.
Will Carrick-Smith and Myles Dorrian crossed for Exeter, with Henry Slade kicking the rest of their points, while Morgan Allen scored Ospreys' only try of the game in the first half.
Three penalties during the final quarter from replacement Aled Thomas saw the Scarlets claim a 22-16 victory over regional rivals Cardiff Blues.
The visitors, who had opened their Anglo-Welsh campaign with a win over Wasps last weekend, led 13-3 during the opening half thanks to a try from flanker Luke Hamilton and eight points from fly-half Gareth Davies.
But the Scarlets hit back through a converted penalty try and a Liam Williams penalty to level. Davies put the Blues back in front, but Thomas stepped up to give the hosts the victory.
- Sports & Recreation