(1) Background: Extravasation injuries in pediatric patients can lead to significant harm if they are not promptly diagnosed and treated. However, evidence-based standardization on extravasation management remains limited, particularly for extensive wound necrosis. This case report presents the management of an 8-week-old premature patient with an extensive extravasation injury involving the right forearm and dorsum of the hand. (2) Methods: The patient was evaluated by a multidisciplinary team in our Neonatal Intensive Care Unit. Surgical intervention involved the debridement of necrotic tissues, followed by temporary coverage with an acellular dermal matrix. Definitive coverage was achieved through Meek micrografting after three weeks. Physical therapy was provided with pre- and post-rehabilitation range of motion assessed using goniometric measurements. Scar quality was evaluated using the Vancouver Scar Scale. (3) Results: The engraftment rate of the Meek micrografts was 93%, with 16 out of 226 micrografts lost. The patient achieved a Vancouver Scar Scale score of 6, indicating a moderate degree of scarring. Significant improvements in elbow, wrist, and finger joint range of motion were observed at a 1-year follow-up. (4) Conclusions: Close observation and heightened awareness of extravasation risks by trained personnel are crucial. Meek micrografting combined with dermal substitute coverage represents an innovative approach to managing extravasation wounds in pediatric patients.