(1) Background: Obstetric brachial plexus palsy (OBPP) is an unpredictable and unpreventable neurological injury, caused by shoulder dystocia during birth, that affects the brachial plexus and leads to motor and sensory deficits in the child's upper extremity. The limited literature on early therapeutic assessment of newborns with OBPP highlights a gap in specialized care that, if filled, could enhance decision-making and support timely treatment. The objective of this paper is to analyze the therapeutic intervention techniques used at an early stage and their functional impact, from the occupational therapy discipline in the treatment of the upper extremity in babies and children with OBPP. (2) Method: Systematic review design and meta-analysis. A systematic review is a comprehensive analysis of existing research on a specific topic, using rigorous methods to identify, evaluate, and synthesize studies. Meta-analysis, often part of a systematic review, combines results from multiple studies to identify overall trends and enhance reliability, providing a clearer summary of evidence. Articles that included pediatric patients (from birth to 12 years of age) with a diagnosis of OBPP were reviewed. The results of the techniques used were analyzed according to each study, with the scale or method of assessment considered by the study for the presentation of data. The articles were assessed for methodological quality using the "PEDro Validity Scale". (3) Results: A total of 2190 articles were found, with 108 analyzed and 22 fully meeting this study's standards. Fourteen had a quantitative design, while the others included clinical guidelines. The most statistically reliable intervention techniques were CIMT (constraint-induced movement therapy) and splinting (dynamic and static), with second-tier techniques like joint manipulation, NMES, early infant management education, and serial casting used when needed. This study focused on children from birth to eight years old, with assessment tools primarily measuring upper limb range of motion, external rotation, supination, and impairment levels, though bimanual activity assessment was less common. (4) Conclusions: The early implementation of the techniques that provide us with the most data are CIMT, splinting, NMES, and joint manipulation linked to health education for families. In second place, we have the use of TB infiltrations and serial casts, when the treatment of the previous techniques fails in some cases.