Caversham was given a free kick and Cook placed himself in front of the Roslyn-Wakari goalkeeper and glanced the ball into the net for a 2-1 win at Ellis Park on Saturday.
That goal has kept coach Steve Fleming's hopes of a Chatham Cup win alive.
His team has reached the semifinals for the past two years.
Caversham will meet Dunedin Technical in the final to find the team that will qualify for the quarterfinals of the national knockout tournament, which was first held in 1923.
Dunedin Technical had too much class and beat Northern 4-0 in the other local derby.
Fleming's team won a hard-fought match that produced the passionate football that is typical of Chatham Cup contests.
Caversham took the lead after 20min when Roslyn defender Michael Eisenhut put the ball into his own net.
The game was played at a fast pace with Caversham dominating the first 30min and creating goal-scoring opportunities.
Roslyn-Wakari defended bravely and had its opportunities on attack but was unlucky to have two goals disallowed.
Roslyn equalised after 15min in the second spell when midfielder Paul Steel scored.
The game was played at a hectic pace after this with both sides seeking the winning goal, which eventually came through Cook.
Aaron Burgess scored three goals to pave the way for a comfortable Tech win over Northern.
He is the top scorer in the premier league this year.
It continued the remarkable goal-scoring record Burgess has had in his long career, with his highlight being the haul of five when he dazzled Waitakere in a 6-1 Tech win in 2000.
Technical is a formidable attacking unit, with Burgess backed by John Lang, who scored the other first-half goal to give their team a 2-0 lead at the break.
Lang was the top scorer in last year's league.
Burgess weaved more of his magic in the second spell to add two more goals.
Technical could have added four more goals but for the brilliance of Northern goalkeeper Ike Malpass, who was at his acrobatic best in front of his goal, which was peppered by Burgess and Lang throughout the game.
Dunedin Technical 4 (Aaron Burgess 3, John Lang), Northern 0; Caversham 2 (Tim Cook, own goal), Roslyn-Wakari 1 (Paul Steel).
