Left-hander Flower, who reached his hundred off just 105 balls, finished unbeaten on 110 while Cook, who scored 120 in the first innings, fell three short of his second century of the match.
Warwickshire captain Nick Knight, who had Flower's old Zimbabwe team-mate Heath Streak leading the Bears' attack, set MCC a target of 296 runs from 62 overs following a sporting lunchtime declaration.
But the champions were without injured South African quick Dewald Pretorius and the tone of MCC's reply was set by batsman-keeper Matt Prior who raced to 29 before being caught behind off Streak.
It was a disappointing end for Prior, who made 70 at more than a run a ball in the first innings, in a match watched by England's chairman of selectors David Graveney.
Prior, who hopes to challenge Geraint Jones for the England wicket-keeping position, said: "I felt in control all through my first innings and even more so in the second and I should have gone on.
"I played some nice shots to get a 29 but a pretty 29 doesn't really help anyone. Hundreds make headlines, not pretty starts."
Looking ahead, he added: "The aim is to get a full season keeping wicket at Sussex under my belt. I've shown people here that I can bat and play. Everyone playing in this match wants to play for England in the future and I am no different.
"I would love to do that more than anything and I will just keep pushing hard."