Tymrak netted just before the hour while Jackson added insurance in injury time to give City all three points in Coffs Harbour.
The result means further heartbreak for the fifth-placed Jets whose finals hopes have evaporated in the final round for the third season in succession.
It is a second-straight tight win for City following on from their recent six-match winless run, but the scrappy nature of the victory suggests the champions have much work to do if they are to reach another grand final.
It was a winner-takes-all contest but the Jets rarely threatened, despite knowing a draw would suit City.
Instead, it was Jets goalkeeper Katelyn Rowland who was kept busy, keeping out several first-half efforts notably from Marianna Tabain and Beverly Yanez.
Relief finally came for City on 58 minutes after fullback Teigen Allen made a driving run down the flank, before teeing up American striker Erica Tymrak who coolly finished past compatriot Rowland.
Substitute Rhali Dobson then fired wide with only the goalkeeper to beat in what was Newcastle's best opening of the match.
Chapman sealed the deal in the dying moments of the game with a shot that went through the hands of the otherwise impeccable Rowland.
City had started the round in fourth, but the win over Newcastle means they could finish as high as second and earn a home semi-final, should both Canberra and Perth Glory lose over the weekend.