So in general this can be achieved setting resource type as Job. Job will be blocking Tiller until it will complete.
There is a small issue here: if job will not complete it will be blocking Helm chart deployment infinite amount of time. To avoid that, need to set spec.activeDeadlineSeconds. It will timeout the job if it will not complete until the set time limit.

apiVersion: batch/v1
kind: Job
metadata:
  name: "{{ .Release.Name }}-database-connection-test"
  annotations:
    "helm.sh/hook": pre-install,test-success
    "helm.sh/hook-delete-policy": before-hook-creation
spec:
  ttlSecondsAfterFinished: 300
  backoffPolicy: 1
  activeDeadlineSeconds: 100
  template:
    spec:
      containers:
      - name: {{ .Release.Name }}-database-connection-test
        image: {{ template "mongo.image" . }}
        imagePullPolicy: Always
        env:
          - name: HOST
            value: {{ .Values.mongo.host }}
          - name: PORT
            value: {{ .Values.mongo.port | quote }}
          - name: DATABASE_NAME
            value: {{ .Values.mongo.databasename }}
          - name: USERNAME
            value: {{ .Values.mongo.username }}
          - name: PASSWORD
            value: {{ .Values.mongo.password }}
        command: ["sh", "-c", "mongo --username $USERNAME --password $PASSWORD --authenticationDatabase $DATABASE_NAME --host $HOST --port $PORT"]
      restartPolicy: Never

It's kind of a workaround, because initially Helm test annotation shouldn't be used alongside other hooks.