Self-testing refers to the phenomenon that certain extremal quantum correlations (almost) uniquely identify the quantum system under consideration. For instance, observing the maximal violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality certifies that the two parties share a singlet. While self-testing results are known for several classes of states, in many cases they are only applicable if the observed statistics are almost perfect, which makes them unsuitable for practical applications. Practically relevant self-testing bounds are much less common and moreover they all result from a single numerical method (with one exception which we discuss in detail). In this work we present a new technique for proving analytic self-testing bounds of practically relevant robustness. We obtain improved bounds for the case of self-testing the singlet using the CHSH inequality (in particular we show that nontrivial fidelity with the singlet can be achieved as long as the violation exceeds β∗=(16+142√)/17≈2.11). In the case of self-testing the tripartite Greenberger-Horne-Zeilinger state using the Mermin inequality, we derive a bound which not only improves on previously known results but turns out to be tight. We discuss other scenarios to which our technique can be immediately applied.