Medical tubular anatomical structures are inherently three-dimensional conduits with lumens, enclosing walls, and complex branching topologies. Accurate reconstruction of their geometry and topology is crucial for applications such as bronchoscopic navigation and cerebral arterial connectivity assessment. Existing methods often rely on voxel-wise overlap measures, which fail to capture topological correctness and completeness. Although topology-aware losses and persistent homology constraints have shown promise, they are usually applied patch-wise and cannot guarantee global preservation or correct geometric errors at inference. To address these limitations, we propose a novel TopoSculpt, a framework for topological refinement of 3D fine-grained tubular structures. TopoSculpt (i) adopts a holistic whole-region modeling strategy to capture full spatial context, (ii) first introduces a Topological Integrity Betti (TIB) constraint that jointly enforces Betti number priors and global integrity, and (iii) employs a curriculum refinement scheme with persistent homology to progressively correct errors from coarse to fine scales. Extensive experiments on challenging pulmonary airway and Circle of Willis datasets demonstrate substantial improvements in both geometry and topology. For instance, $β_{0}$ errors are reduced from 69.00 to 3.40 on the airway dataset and from 1.65 to 0.30 on the CoW dataset, with Tree length detected and branch detected rates improving by nearly 10\%. These results highlight the effectiveness of TopoSculpt in correcting critical topological errors and advancing the high-fidelity modeling of complex 3D tubular anatomy. The project homepage is available at: https://github.com/Puzzled-Hui/TopoSculpt.