Captain Andrew Strauss, Kevin Pietersen and Paul Collingwood all fell within touching distance of three figures, but they still managed to steer England to 414 all out in response to the hosts' 574 for eight declared.
Batting a second time, Windies A had reached 16 without loss when the match was called off at 4:01 p.m., with no outright outcome possible.
The left-handed Strauss, un-beaten on 59 overnight, made 97 while Pietersen converted his 12 at the start of the day into 90. Collingwood struck 82.
Windies A's bowlers stuck to their task manfully in the face of England's powerful batting line-up and were rewarded.
Fast bowler Kevin McClean collected three for 56, including the scalps of Collingwood and Cook for a first-ball duck on the previous day, while leg-spinner Gavin Wallace finished with three for 149, prising out Strauss three runs short of his century.
aggressive
Off-spinner Amit Jaggernauth, chosen in the 14-man squad for the first Test in Jamaica on Wednesday, claimed two for 100.
Resuming at 134 for two, England got a 132-run stand for the third wicket between Strauss and Pietersen to propel them in the first session.
Strauss hit 16 fours and one six off 135 balls in 196 minutes at the crease while the aggressive right-handed Pietersen smashed 17 fours off a mere 82 balls.
The England skipper was bowled by Wallace when seemingly set for a century and Pietersen followed him soon afterward, becoming the first of Jaggernauth's wickets when he was also bowled over.
When Owais Shah was trapped lbw by Jaggernauth, England had lost three wickets for 28 runs to slip to 270 for five.
Collingwood, whose knock consumed 122 balls and included five fours and three sixes, added 80 for the sixth wicket with Matt Prior (36) as England fought back.
When they were separated in the post-lunch session, the innings drifted aimlessly but it was time enough for the Windies bowlers to improve their figures.