The birth of a child diagnosed with autism spectrum disorder (ASD) tends to strongly disrupt family dynamics and functioning. However, the severity of the impact may be softened if the family feels supported during the diagnostic process. The Valencia region (Spain)—where this study is located—recently put in place a protocol to improve ASD detection and support for families. The aim of this study was to identify these families’ views on the quality of the process experienced and the operation of the new system. The participants were 34 families with a child who had been diagnosed with this condition. A mixed methodological approach was adopted. A descriptive analysis and an interpretative-phenomenological study were performed using SPSS v. 25 and AQUAD 7, respectively. The results showed that while families seemed to be relatively satisfied, there is still room for improvement in some important areas such as guidance and emotional support, the training of the professionals involved, and waiting time. It was concluded that significant improvements must be implemented in the new diagnostic model to successfully meet the demands of families in a context characterized by a paucity of studies.