This generalization of Tic-Tac-Toe is called m,n,k-game. (the goal is to get k in a row on a (m,n) board).
Some known bounds: (source wikipedia)
(5,5,4) is a draw.
(6,6,5) is a draw.
(7,7,5) and (8,8,5) are draws.
(15,15,5) is a win.
(9,6,6) and (7,7,6) are both draws via pairings.
When the goal is 9 or larger (k>=9) the second player can force a ...