As we already know, ports up until 1024 are only accessible by root processes. Therefore, in order to be able to run any process on the ports below 1024 the following command should be run.
sudo setcap cap_net_bind_service=+ep /path/to/application
Keep in mind that these restrictions are in place for a reason, and having a reverse proxy would be a better solution here. However there are some extreme cases when this is actually needed.