An iron-binding mung bean protein hydrolysate (MBPH) was prepared using a continuous enzymatic membrane reactor followed by peptide separation on anion-exchange (AEC) and reverse-phase HPLC (RP-HPLC) columns. Amino acid sequences of peptides present in the RP-HPLC fraction with the strongest iron-binding capacity were identified using mass spectrometry, and ten peptides of 5–8 amino acids synthesized for antioxidant characterization. Five fractions (AF1– AF5) with higher iron-binding capacity (88.86 ± 6.43 to 153.59 ± 2.18 mg/g peptide) when compared to the MBPH (36.81 ± 0.93 mg/g peptide) were obtained from AEC. PAIDL had the significantly (p < 0.05) highest iron-binding capacity, but LLLLG and LLGIL showed the strongest metal chelating activity. However, PAIDL (46.63%) and LLGIL (81.27%) had significantly (p < 0.05) better DPPH radical scavenging activity than the other peptides. PAIDL and LLGIL were also the most effective (p < 0.05) hydroxyl radical neutralizers with an effective concentration that scavenged 50% (EC50) values of 0.09 and 0.37 mM, respectively. PAIDL and AIVIL showed the lowest EC50 values of 0.07 mM each for superoxide radical scavenging activity. We conclude that short chain length in combination with leucine as the C-terminal amino acid residue contributed to the strong antioxidant properties of peptides in this study.