Yes, REST API is the easiest way.
You need to perform a Build Promotion call. Please note it requires Artifactory Pro.
It's a POST request, accepting simple json string, in which only two properties are mandatory: status and ciUser.
The call should look something like this:
curl -X POST -u admin:password -H "Content-Type: application/json" -d '{"status":"tests passed","ciUser":"jenkinsAdmin"}' "http://localhost:8081/artifactory/api/build/promote/buildName/buildNumber"