The presence of dust at high redshift requires efficient condensation of grains in supernova (SN) ejecta, in accordance with current theoretical models. Yet observations of the few well-studied supernovae (SNe) and supernova remnants (SNRs) imply condensation efficiencies which are about two orders of magnitude smaller. Motivated by this tension, we have (i) revisited the model of Todini & Ferrara for dust formation in the ejecta of core collapse SNe, and (ii) followed, for the first time, the evolution of newly condensed grains from the time of formation to their survival – through the passage of the reverse shock – in the SNR. We find that 0.1–0.6 M⊙ of dust form in the ejecta of 12–40 M⊙ stellar progenitors. Depending on the density of the surrounding interstellar medium, between 2 and 20 per cent of the initial dust mass survives the passage of the reverse shock, on time-scales of about 4–8 × 104 yr from the stellar explosion. Sputtering by the hot gas induces a shift of the dust size distribution towards smaller grains. The resulting dust extinction curve shows a good agreement with that derived by observations of a reddened QSO at z= 6.2. Stochastic heating of small grains leads to a wide distribution of dust temperatures. This supports the idea that large amounts (∼0.1 M⊙) of cold dust (T∼ 40 K) can be present in SNRs, without being in conflict with the observed infrared emission.