Abstract We consider the first-order system space–time formulation of the heat equation introduced by Bochev and Gunzburger (in: Bochev and Gunzburger (eds) Applied mathematical sciences, vol 166, Springer, New York, 2009), and analyzed by Führer and Karkulik (Comput Math Appl 92:27–36, 2021) and Gantner and Stevenson (ESAIM Math Model Numer Anal 55(1):283–299 2021), with solution components $(u_1,\textbf{u}_2)=(u,-\nabla _\textbf{x} u)$. The corresponding operator is boundedly invertible between a Hilbert space U and a Cartesian product of $L_2$-type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides $L_2$-norms of $\nabla _\textbf{x} u_1$ and $\textbf{u}_2$ , the (graph) norm of U contains the $L_2$-norm of $\partial _t u_1 +{{\,\textrm{div}\,}}_\textbf{x} \textbf{u}_2$. When applying standard finite elements w.r.t. simplicial partitions of the space–time cylinder, estimates of the approximation error w.r.t. the latter norm require higher-order smoothness of $\textbf{u}_2$. In experiments for both uniform and adaptively refined partitions, this manifested itself in disappointingly low convergence rates for non-smooth solutions u . In this paper, we construct finite element spaces w.r.t. prismatic partitions. They come with a quasi-interpolant that satisfies a near commuting diagram in the sense that, apart from some harmless term, the aforementioned error depends exclusively on the smoothness of $\partial _t u_1 +{{\,\textrm{div}\,}}_\textbf{x} \textbf{u}_2$, i.e., of the forcing term $f=(\partial _t-\Delta _x)u$ . Numerical results show significantly improved convergence rates.