There's actually more than one strategy you could employ for this. Three, in actual fact, depending on where you start on the board.
It's easier to find a strategy assuming Fluttershy will not be playing optimally, because otherwise constantly second-guessing could lead you to a loss. Now, obviously, as you can't see Fluttershy's moves, you're going to have to lock her into blocking your plays to make her moves more predictable.
Let's label the squares from 1-9, as follows:
1 2 3
4 5 6
7 8 9
The first strategy can be summarised as "start with 5, then move down the left-most branch on the diagram below. If you're blocked, move to the next branch along and so on". Coloured circles below represent draws, all others represent wins:
Here's a second strategy where you start with the corner square 1, following the same rules as the first diagram:
Finally, here's another strategy where you start with square 2, and then follow the steps going right, moving down if you're blocked:
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I gotta say, your diagram is much easier to follow.