We study the maximum constraint satisfaction problem, Max-CSP, in the streaming setting. Given $n$ variables, the constraints arrive sequentially in an arbitrary order, with each constraint involving only a small subset of the variables. The objective is to approximate the maximum fraction of constraints that can be satisfied by an optimal assignment in a single pass. The problem admits a trivial near-optimal solution with $O(n)$ space, so the major open problem in the literature has been the best approximation achievable when limiting the space to $o(n)$. The answer to the question above depends heavily on the CSP instance at hand. The integrality gap $α$ of an LP relaxation, known as the BasicLP, plays a central role. In particular, a major conjecture of the area is that in the single-pass streaming setting, for any fixed $\varepsilon > 0$, (i) an $(α-\varepsilon)$-approximation can be achieved with $o(n)$ space, and (ii) any $(α+\varepsilon)$-approximation requires $Ω(n)$ space. In this work, we fully resolve the first side of the conjecture by proving that an $(α- \varepsilon)$-approximation of Max-CSP can indeed be achieved using $n^{1-Ω_\varepsilon(1)}$ space and in a single pass. Given that Max-DiCut is a special case of Max-CSP, our algorithm fully recovers the recent result of [ABFS26, STOC'26] via a completely different algorithm and proof. On a technical level, our algorithm simulates a suitable local algorithm on a reduced graph using a technique that we call *two-tier sampling*: the algorithm combines both edge sampling and vertex sampling to handle high- and low-degree vertices at the same time.