It is definitely possible to use Content Builder emails in triggered sends as I'm currently doing this. When I go to Email Studio -> Interactions -> Triggered Emails and click on Create, in the "Content"-section I can select either Classic Emails or Content Builder Emails (see screenshot below).
If you don't have the possibility to select Content Builder Emails I guess this requires opening a support case - like so many feature activations in Marketing Cloud do.
As you added in your comment, you are using triggered sends via Markting Cloud Connect (which can be triggered on object creation or update).
With the June 2017 release, Salesforce added support for Content Builder E-Mails using Marketing Cloud Connect. However, this isn't available for current customers without manually updating Marketing Cloud Connect to version 208:
The new email selection is available to Email Sends and Triggered Sends. This release does not support using Content Builder emails with AB Tests.
This feature is available to any clients utilizing the 208 version of the package.
Workaround using Journey Builder (doesn't require Connector Update):
However you could create a journey using Journey Builder in Marketing Cloud. This journey needs to be triggered by a Salesforce Data event with the same criteria you would define in Sales/Service Cloud when configuring your triggered send. Your journey would then be very basic as the only activity added to the canvas would be an email send activity, where you can use Content Builder emails. Note that Salesforce Data Events are not subject to the synchronization schedule you define for your data sources and are triggered immediately, so this isn't an issue.
Documentation articles related to the Journey Builder workaround: