To solve the equation $x^{2}+2x-1=0$ using the completing the square method, we follow these steps:

1. Start with the given equation:
$$x^{2}+2x-1=0$$

2. Move the constant term to the other side:
$$x^{2}+2x = 1$$

3. To complete the square, we need to add $(\frac{2}{2})^{2} = 1$ to both sides of the equation. This is because the coefficient of $x$ is $2$, and we use the formula $\left(\frac{\text{coefficient of } x}{2}\right)^{2}$:
$$x^{2}+2x+1 = 1+1$$

4. Simplify the right side and write the left side as a square:
$$x^{2}+2x+1 = 2$$
$$\left(x+1\right)^{2} = 2$$

Therefore, the equation in the form of completing the square is $\boxed{(x+1)^{2}=2}$, which corresponds to choice $\boxed{B}$.