In the paper, an original design of an ozone-catalytic device, in which the process of ozone formation occurs directly inside the catalyst monolith of the honeycomb structure, is proposed for solving the problem of exhaust gases purification at the "cold start" of an automobile engine. For the considered device, various regimes of operation were examined, and the equations allowing us to calculate the concentrations of ozone depending on time were obtained. The developed mathematical model shows a significant decrease in ozone concentration after the "cold start" is over and the temperature of the exhaust gases increases, because of changes in the balance of various chemical reactions. Thus, the ozone concentration is maximum during a cold start, precisely when the lower temperatures do not allow other reactions than those of oxidation of toxic gases with ozone, and decreases by the time the catalytic unit warms up, when effective neutralization of toxic impurities becomes possible without the participation of ozone. The results of the mathematical modeling were confirmed by testing this device for purifying exhaust gases of a YaMZ-238M2 engine. The test results show that the use of ozone-catalytic reactions significantly increases the efficiency of neutralizing toxic impurities under cold start conditions, compared to the case when only catalytic reactions with oxygen without the participation of ozone are used.