A novel multifunctional isocyanate curing agent, denoted as TCI, was facilely synthesized via a one-step reaction involving 1,3,5-tris(2-hydroxyethyl)cyanuric acid and hexamethylene diisocyanate. The structural design of TCI incorporates a rigid triazine ring core and three additional urethane linkages, enabling the construction of high-performance crosslinked networks within glycidyl azide polymer (GAP)-based energetic elastomers. By systematically comparing TCI with the commercially available curing agent N100, the structure–property relationships were elucidated through a combination of curing kinetics, spectroscopic characterization, mechanical testing, and thermal analysis. TCI exhibits superior reactivity toward GAP, effectively compensating for the low reactivity of secondary hydroxyl groups. Structural characterization via XRD, SAXS, and FTIR demonstrated that TCI facilitates the formation of dense and stable hydrogen-bonding networks, which reduce the intermolecular chain spacing (0.424 nm for TCI–GAP vs. 0.436 nm for N100–GAP) and optimize the microphase separation behavior of the elastomer networks.Mechanical testing demonstrated that the tensile strength and elongation at break of TCI–GAP reach 1.74 MPa and 297.7%, respectively, representing increases of 95.5% and 163% over N100–GAP (0.85 MPa and 105.8%). Morphological analysis confirmed the presence of uniformly distributed crosslinking junctions and ductile fracture features in TCI–GAP, which are responsible for the improved load-bearing capacity and energy dissipation efficiency. Dynamic hydrogen-bonding interactions were identified as the key mechanism governing the mechanical and thermal performance of TCI-GAP. DMA and LF-NMR results indicated that TCI–GAP possesses higher storage modulus and more restricted segmental motion, leading to excellent thermomechanical stability. The novel TCI curing agent offers a facile and effective approach to simultaneously improve the mechanical strength, toughness, and thermal stability of GAP-based networks, showing great promise for applications in advanced solid propellants and energetic materials.