Context. Classical novae (CNe) have recently been reported to represent the major class of supersoft X-ray sources (SSSs) in the central region of our neighbour galaxy M 31. Aims: We carried out a dedicated monitoring of the M 31 central region with XMM-Newton and Chandra in order to find SSS counterparts of CNe, determine the duration of their SSS phase and derive physical outburst parameters. Methods: We systematically searched our data for X-ray counterparts of CNe and determined their X-ray light curves and spectral properties. Additionally, we determined luminosity upper limits for all novae from previous studies which are not detected anymore and for all CNe in our field of view with optical outbursts between May 2005 and March 2007. Results: We detected eight X-ray counterparts of CNe in M 31, four of which were not previously known. Seven sources can be classified as SSSs, one is a candidate SSS. Two SSSs are still visible more than nine years after the nova outburst, whereas two other nova counterparts show a short SSS phase of less than 150 days. Of the latter sources, M31N 2006-04a exhibits a short-time variable X-ray light curve with an apparent period of (1.6±0.3) h. This periodicity could indicate the binary period of the system. There is no X-ray detection for 23 out of 25 CNe which were within the field of view of our observations and had their outburst from about one year before the start of the monitoring until its end. From the 14 SSS nova counterparts known from previous studies, ten are not detected anymore. Additionally, we found four SSSs in our XMM-Newton data without a nova counterpart, one of which is a new source. Conclusions: Out of eleven SSSs detected in our monitoring, seven are counterparts of CNe. We therefore confirm the earlier finding that CNe are the major class of SSSs in the central region of M 31. We use the measured SSS turn-on and turn-off times to estimate the mass ejected in the nova outburst and the mass burned on the white dwarf. Classical novae with short SSS phases seem to be an important contributor to the overall population. Partly based on observations with XMM-Newton, an ESA Science Mission with instruments and contributions directly funded by ESA Member States and NASA.