Despite Will Fletcher pulling back a goal late on, the visitors held on for all three points.
Jarvis's first half goal proved the difference between the sides.
A 99th minute penalty from Rice meant the match finished all square.
Goals from Carmichael, Burrows and Jackson ensured the Magpies start 2025 with a win.
Jackson’s 90+5′ minute header rescued a point for the Magpies.
Billy Walker's first half goal gave the Magpies their second away win of the season in Tim Sills 100th game in charge.
In a reverse of the scoreline from August, Winchester took the points.
Brown's brilliant free kick completed Basingstoke's second half turnaround.