Simple Summary It becomes critical to understand the effects that the current pandemic can have on the welfare of household dogs and cats, in order to develop programs that can support the owners in the care of their companion animals during such a critical time. The current survey showed that the constant presence of the owners at home during the lockdown favored the decrease in most of the behavioral problems considered. For example, during the lockdown period, dogs were more willing to play inside the house, while cats were more social and had a better appetite. In addition, litter box management for cats was improved during the lockdown, which might explain the decrease in the frequency of inappropriate elimination. Only anxiety-related behaviors in dogs increased during the lockdown, and these were associated with a reduction in play activity and altered sleeping patterns. On the other hand, most of the behavioral issues considered in the study were more frequent during the pre-lockdown period, which might have been associated with the long hours that pets spent alone at home, with reduced possibilities of interaction with their owners. Therefore, it is critical to develop support programs for pet owners, for both emergency situations such as lockdowns and normal times, to allow the establishment and maintenance of a healthy human-pet relationship and good pet welfare. Considering the effects that the COVID-19 pandemic had and still has on human psychological health, it is expected that it might also affect household dogs' and cats' welfare. The current study explores the behavioral changes in dogs and cats before (BL) and during the lockdown (DL), as reported by their owners in China. Besides demographic parameters, variables related to the daily management of dogs and cats were analyzed in relation to behavioral problems, stress-related behaviors, and anxiety-related behaviors before and during the lockdown. A total of 261 questionnaires were collected. In general, behavioral problems and stress-related behaviors in dogs (p < 0.001) and cats (p < 0.001) decreased DL compared to BL, while anxiety-related behaviors in cats did not show any differences between the two periods considered. On the other hand, anxiety-related behaviors were more frequent in dogs DL (36.3%) compared to BL (35%), which were associated with reduced frequency of play activities with the owners (p = 0.016) and altered sleeping habits (p < 0.01). During the lockdown, dogs' and cats' daily routines and management (feeding and sleeping habits, dogs' walks, dogs' and cats' play activities, litter box management, and cats' lifestyle) experienced changes, but they were not associated with any behavioral issues. On the other hand, the behavioral issues considered for dogs and cats were more frequent BL, which were influenced by the daily management of the pets. The current study showed how critical the attention the owners can provide to the pets could be, to improve their companion animals' welfare. Therefore, it is important to provide pet owners with behavioral management support both during particularly difficult periods such as a lockdown and during regular daily routines.