As cloud services are growing and generating high revenues, the cost of downtime in these services is becoming significantly expensive. To reduce loss and service downtime, a critical primary step is to execute incident triage, the process of assigning a service incident to the correct responsible team, in a timely manner. An incorrect assignment risks additional incident reroutings and increases its time to mitigate by 10x. However, automated incident triage in large cloud services faces many challenges: (1) a highly imbalanced incident distribution from a large number of teams, (2) wide variety in formats of input data or data sources, (3) scaling to meet production-grade requirements, and (4) gaining engineers' trust in using machine learning recommendations. To address these challenges, we introduce DeepTriage, an intelligent incident transfer service combining multiple machine learning techniques - gradient boosted classifiers, clustering methods, and deep neural networks - in an ensemble to recommend the responsible team to triage an incident. Experimental results on real incidents in Microsoft Azure show that our service achieves 82.9% F1 score. For highly impacted incidents, DeepTriage achieves F1 score from 76.3% -- 91.3%. We have applied best practices and state-of-the-art frameworks to scale DeepTriage to handle incident routing for all cloud services. DeepTriage has been deployed in Azure since October 2017 and is used by thousands of teams daily.