The HDInsight service has a REST API called WebHCat (formerly Templeton, so you might see that in the docs) which you can use from node easily enough.
The documentation is here http://hive.apache.org/docs/hcat_r0.5.0/rest.html
To just submit a job, the call you're looking for is:
https://mycluster.azurehdinsight.net/templeton/v1/hive
You just post it a query in the execute parameter (form encoded). There are a few other options to do with status location, and completion callbacks which will maybe useful in your application, all documented at http://hive.apache.org/docs/hcat_r0.5.0/hive.html
Note that the templeton instance in HDInsight runs on the standard ssl port (443) rather than the templeton default used in other distributions (50111)