Bell smashed the home attack for 187 from 145 balls ??? the highest by an England batsman in limited overs cricket ??? as the tourists set a daunting target of 392 from their 50 overs.
The score also matched England???s previous record total, set in 2005 against Bangladesh.
But Maxwell wasn???t fazed, scoring 136 from 91 balls, although only one other batsman passed 20 for the Prime Minister’s XI as England won despite some nervy moments.
Moeen Ali and Bell, who had impressed in the opening game of the tour on Monday, did so again with a blistering start to the game.
Ali hit 13 fours in a sparkling 49-ball 71, and when he eventually fell to the spin of Maxwell, England had already put on 113 inside 15 overs.
The scoring pace showed no sign of slowing when James Taylor came in as he and Bell pushed and prodded the struggling attack with a little more finesse than Ali had before his departure.
Taylor matched his score of 71, although his knock came off 77 balls and included only four boundaries. He would eventually fall to left-armer Jason Behrendorff (4-79) after a second-wicket partnership of 141.
Joe Root managed only 13 before falling to David Moody and Jos Buttler added a quickfire 25 off 14 deliveries, smashing two fours and to sixes as England looked to reach 400.
Behrendorff would deny them the chance to reach that landmark, picking up the wickets of Buttler, Eoin Morgan (seven) and Bell in the final over as England finished on 391-6.
In reply, Cameron Bancroft (15) tried to kick-start the run-chase with an early six off Chris Woakes but was gone two balls later when the all-rounder snuck one through his defences and clipped the off stump.
Chris Jordan also took an early wicket, that of captain Chris Rogers for a boundary-less 20. But the new batsman at the other end, Maxwell, took a greater liking to him, smashing 21 runs from one over which included three fours and an effortless six over fine leg.
Maxwell raced to 50 in 32 balls, continuing to counter-attack with back-to-back boundaries off James Tredwell. His hundred soon followed, coming up in 35 fewer deliveries than Bell’s. Suddenly an England win looked far from straight-forward.
Their hosts bettered England???s 25-over total by 18 runs, albeit three wickets down rather than one ??? Peter Hanscomb (17) the other man out, bowled by Ali.
But the batting powerplay, called in the 28th over, proved to be the decisive moment of the run-chase as an over later Maxwell struck Stuart Broad (4-40) ??? England???s pick of the bowlers ??? for three successive boundaries before finally succumbing, caught at gully off a thick edge.
It left his side needing a further 176 from 126 balls, a task that proved beyond the lower-order despite some useful contributions from Jake Doran (37) and Lega Siaka (27), Pat Cummins (27).
Not wholly convincing ??? particularly the bowling ??? but England eventually secured the win, bowling the home side out for 331 in the 49th over.