Moeen scored 128 and shared 172 for the first wicket with the stuttering Ian Bell, who contributed 54.
But, when they were parted, England could only manage 131 runs in the final 20 overs as they posted 303-8.
Still, it was too many for Scotland, who never threatened an upset, despite Kyle Coetzer's 71, and were bowled out for 184, with Steven Finn taking 3-26.
Following heavy defeats against co-hosts Australia and New Zealand, the success in Christchurch provides England with a much-needed morale boost.
However, little can be learned as to whether England are in better shape for the sterner tests to come - starting with Sri Lanka in Wellington on Sunday - given the nature of the opposition.
scoreboard:
| England 303-8 (50 overs): Moeen 128, Bell 54, Davey 4-68 |
|
Scotland 184 (42.2 overs): Coetzer 71, Finn 3-26 |
| England won by 119 runs abd71 |
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