There are a few possibilities to resolve this:
Azure VNET based point-to-site service can be used to connect one on-premise DB to an Azure-hosted application. This is valid where there are limited resources to be connected via VPN.
In case of more resources for connection, site-to-site VPN or express routes are the solutions. Site-to-site might cause network latency as the VPN works only via public infrastructure, which is the Internet. In that case, express routes can be used as it has a dedicated leased line that solves latency issues.
In case VNET is not preferred, Windows Communication Foundation (WCF) service can be developed and hosted on-premise. It will have CRUD operations intended solely for the database that is hosted on-premise. It uses the service bus relay that can build communication between the Azure-hosted app to the WCF service for database access.