Moeen ensured any jitters following Friday’s Wellington humiliation against New Zealand were quickly eased as he cracked 128 from 107 balls and shared in a record opening stand of 172 with Ian Bell.
From there England were always in control of the potential banana-skin contest, even though they did not fully exploit the foundation laid by the openers in reaching 303-8.
It was a target that proved beyond Scotland, still searching for a first World Cup win in their third visit to the tournament, as they were bowled out for 184 in 42.2 overs.
Steven Finn bounced back from his Brendon McCullum mauling - when his two overs against New Zealand cost 49 - by taking 3-26 while Moeen followed his century with figures of 2-47.
Kyle Coetzer top-scored with 71 for Scotland after Josh Davey took 4-68.
Thumping back-to-back defeats against co-hosts Australia and New Zealand had left England with little room for error in their remaining pool games but they will now head back to Wellington, for their clash with Sri Lanka next Sunday, with the first part of a recovery mission to reach the quarter-finals complete.
Eoin Morgan will also feel as though he has started to put his horror run of outs behind him after hitting 46 from 42 balls.
The captain had initially taken 10 balls to get off the mark as England lost direction around the batting powerplay, losing 3-2 at one stage, but found enough fluency to drag his team beyond a score of 300.
That had appeared the minimum standard when Moeen and Bell coasted to the England record opening partnership at a World Cup - beating the previous mark of 158 set by Dennis Amiss and Barry Wood in 1975.
Further records appeared ready to tumble as Moeen took on the aggressor’s role and fittingly brought up his century, from 91 balls, when he clubbed his fourth six off spinner Majid Haq.
Bell had crept to his half-century only moments earlier, using only 11 balls less than Moeen needed for his ton, but with the scoreboard ticking over the “clinical” performance the right-hander demanded on the eve of the match was being demonstrated.
But Moeen’s exit in the 35th over prompted a stall as only Morgan, and Jos Buttler with a quickfire 24, were able to strike the ball with the same assurance as their left-handed opener.
It hardly mattered in terms of the result as England’s bowlers ensured Scotland never threatened to pull off an unlikely chase.
Coetzer ensured England were kept honest, but when he and captain Preston Mommsen fell in successive overs, after a 60-run partnership for the fourth wicket, Scotland’s faint hopes were extinguished.
Moeen was the bowler to remove Coetzer, who was out of favour at Northamptonshire at the end of the summer, when he lofted straight to Chris Woakes.
Finn then induced a couple of edges, from Matthew Cross and Davey, before Woakes finished matter when Haq top-edged a pull to Gary Ballance at fine-leg.
England had stuck with the same XI that were crushed by New Zealand as those involved were given the chance at redemption.
Scotland captain Preston Mommsen sent England in on a cloudy morning and with their skittling for 123 in Wellington fresh in the memory.
Moeen survived an early chance on seven, when Freddie Coleman was late on getting down to a chance at cover, but from then on the England openers were hardly troubled.
Moeen responded to his reprieve by thumping Josh Davey for six later in the over and he welcomed Haq into the attack with another towering blow en-route to a 39-ball half-century.
Bell was setting himself to bat through the innings - after England were bowled out in their opening two games of the tournament - and it was a shock when he picked out short cover on 54 when Richie Berrington returned to the attack.
Moeen continued the clatter along, passing his previous ODI best of 119, before he lofted Haq to Coleman on the mid-wicket rope.
It set off the mini-collapse of three wickets in as many overs.
Ballance chopped on from the first ball of the fielding restrictions and Joe Root quickly followed as he nibbled at Josh Davey delivery that seamed away.
Scotland were edging back after restricting England to 22-2 in the powerplay and James Taylor’s 17 needed 26 balls before he was expertly stumped by Matthew Cross.
Buttler was therefore left to come in with only 31 balls left in the innings, hitting some sweet shots in his brief cameo, and with Morgan finding his range England scrambled past 300.