The St Lucian seamer bagged 5-41 to take his seasonal tally to 15 and put himself in a three-way tie for the Division 3 leading wicket-taker award.
Chicot took four of the first five wickets to fall as Purbrook slumped to 52-5 after being inserted.
Then, after skipper Brad Mengham (36) and Alex Cox (28) had added 68 for the sixth wicket, the last five home wickets fell for the addition of just 10 runs as they were dismissed for 130.
Jubin Karippai (2-7 off five overs) and Jack Richards (2-29) were also among the wickets.
Mark Toogood then dealt mainly in boundaries as Gosport raced to a seven-wicket win.
Opening the innings, Toogood thumped 14 fours - hitting 56 of his 62 runs in boundaries and sharing a 99-run stand for the first wicket with Greg Kitchin (36).
Hamid Khan took 2-40 off six overs as Gosport won off the final ball of the 22nd over.
As a result, the situation for Purbrook is already looking tough. They are stranded at the foot of the table, 47 points adrift of third bottom Trojans - and the bottom two are relegated to the Hampshire League.
As for Chicot, he is now level with Sway’s Hugh Bernard and South Wilts 2nds’ James Brewer on 15 wickets.
Havant 2nds opener Andrew Galliers hit a half-century as his side chased 239 to win at South Wilts 2nds.
Galliers struck 67 to help the visitors reach 121-2. After that, middle order pair Oliver Jones (43) and Harrison Barnard (35) helped take Havant to a four-wicket win with six balls remaining.
Earlier, Warren Turner (3-69) - the seventh and final bowler used - had been Havant’s leading wicket-taker as the hosts posted 225-6 (Joe Cranch 56, Nick Patridge 35 not out).
Elsewhere in the division, middle order bat Rahul Bhome helped Trojans sneak past Paultons.
After Paultons were dismissed for 127, Trojans were in the mire on 82-7 (Harman Chandhoke 5-53). But Bhome, batting at No 6, struck an unbeaten 51 as his side won by two wickets.