Dovetail Software leveraged Twilio's telephony web service to integrate their CRM with phone calls, allowing on-call support agents to receive notifications about new urgent cases through a simple and cost-effective solution. They utilized Twilio's Calls Rest API to initiate outbound calls, passing the caller ID and conversation script URL to control the call flow. The implementation was completed within 3 hours, showcasing Twilio's ease of use and developer-friendly resources.