Aussie paceman Magoffin struck first in the eighth over, RR Roy sending a simple catch down the throat of Anyon at mid on, and two overs later Magoffin struck again when Mark Ramprakash edged behind.
After a brief rain delay Anyon got in on the act with two wickets in one over. Zander de Bruyn edged to Mike Yardy at second slip and then former Sussex player Rory Hamilton-Brown got himself out first ball, nicking one down the leg side for Ben Brown to pouch behind the stumps.
A superb diving catch from Luke Wells in the slips accounted for Davies, giving Magoffin a third wicket, and after a quick-fire 17 Maynard prodded behind to Yardy as Anyon also made it three.
From 52/6 Surrey moved on to 78/6 at lunch but with conditions perfect for the Sussex swing bowlers and catches being held the hosts will hope to bowl their neighbours out before the end of the first day.