The 2D directed spanning forest (DSF) introduced by Baccelli and Bordenave is a planar directed forest whose vertex set is given by a homogeneous Poisson point process (PPP), ${\cal N}$, on $\mathbb{R}^2$. If the DSF has direction $-e_y$, the ancestor $h(\mathbf{u})$ of a vertex $\mathbf{u} \in {\cal N}$ is the nearest Poisson point (in the $L_2$ distance) having strictly larger $y$-coordinate. We show that the collection of rescaled DSF paths converges in distribution to the Brownian web (BW) verifying a conjecture by Baccelli and Bordenave (2007). A key ingredient for the proof is to control the tail distribution of the coalescence time between two paths of the DSF. The facts that the DSF spans on a PPP and that its construction is based on the $L^2$ distance-- which is very natural --destroys all Markov and martingale properties on which the existing literature usually relies for proving convergence of directed forests to the BW. Our proof relies on the construction of clever regeneration events exploiting the particular geometry of the DSF. The distance between two given paths (actually two regenerated paths associated to the original ones) considered at the corresponding regeneration times satisfies a certain Markov property w.r.t. an enlarged filtration, and its hitting time of zero is controlled by a new Laplace criterion. We then introduce a new criterion for the convergence to the BW and its dual, inspired from the wedge condition of Schertzer et al. (2017), and show that it is verified. We believe that the ideas in this work can be applied to the convergence to the BW of a large variety of directed forests with intricate dependencies... Finally, the coalescence time estimate for DSF paths is used to quantify the number of semi-infinite paths of the Radial Spanning Tree crossing the circle centred at the origin and with radius $r$: it is negligible w.r.t. $r^{3/4+\varepsilon}$.