James Benning (43) and Scott Newman (51) got Surrey off to a flying start with a stand of 81 from 12 overs as Collymore, signed as a Kolpak player on a one-year deal, conceded 42 runs in his first six overs.
Afzaal and Ramprakash then took up the attack and were particularly harsh on off-spinner Ollie Rayner.
Brown took over from Ramprakash in a profitable stand at the end of their 50 overs, but left-hander Afzaal, who joined the county from Nottinghamshire in the summer, was the star and he reached his century off 92 balls.
The former England batsman hit 12 fours and two sixes before stumps brought an end to Sussex's misery in the field.
Sussex skipper Chris Adams (40), Murray Goodwin (38) and Michael Yardy (39) were unable to get the big score needed and the hosts lost wickets at regular intervals as Surrey sealed an emphatic victory.