A first determination of the strong coupling $\alpha_s(m_Z)$ in next-to-next-to-leading order (NNLO) from inclusive jet and dijet production cross sections in deep-inelastic scattering at HERA is presented. Data collected by the H1 experiment in the years 1995 to 2007 covering the range of momentum transfer $5.5<Q^2<15\,000\,{\rm GeV}^2$ and jet transverse momenta $P_T>4.5\,{\rm GeV}$ are explored. The strong coupling is determined in a fit to inclusive jet and dijet data to $\alpha_s(m_Z)=0.1157\,(6)_{\rm exp}\,(^{+31}_{-26})_{\rm theo}$. Further studies on the phenomenological application of the new NNLO calculations and on fits to the individual data sets are presented. The running of the strong coupling is probed in a single experiment over one order of magnitude in the remormalisation scale and consistency with the QCD expectations is found.