One graph searching technique you can use for solving a Sudoku puzzle is backtracking. Suppose you are given a partially filled Sudoku puzzle. Choose some empty cell, and assign 1 to it. Check if there are any conflicts. If so, erase that 1 you just put in, and put in 2. Keep repeating this until you find a number that causes no conflicts. Now repeat this procedure until the puzzle is solved. It is not hard to see you will eventually find a solution using this strategy, because it will in the worst case try all possible ways filling the puzzle.
And when you think of it, the Sudoku puzzle actually is about getting from node $A$ to node $B$. In this setting, your node A is the partially filled puzzle given to you initially. Node $B$ is a completely filled puzzle with no conflicts. Choosing an empty cell and assign a value to it (corresponding to the situation described in the first paragraph), means you are moving from $A$ to $A_2$, where $A_2$ is a partially solved puzzle. Moving along from here will hopefully make you eventually find $B$, which is a goal state. In this way, you can indeed see the puzzle solving process as a directed graph.
You can see backtracking in action here.