The styrylpyrone dehydrogoniothalamin (1) and two of its dimers (2 and 3) were isolated from the leaves of Aniba heringeri (Lauraceae). Compound 3 is new, while 1 and 2 are being reported for the first time in this species. Structures were determined by 1D- and 2D-NMR spectroscopy, mass spectrometry, and optical rotation data. Cytotoxic effects and selectivity indices were evaluated in five neoplastic cell lines—PC-3 (prostate), 786–0 (renal), HT-29 (colon), MCF-7, and MDA-MB-231 (breast)—and a non-neoplastic cell line, (NIH/3T3, murine fibroblast). Compound 1 inhibited cell growth by 50% (GI50) at concentrations in the 90.4–175.7 μM range, while 2 proved active against MCF-7 and MDA-MB-231 breast cells (GI50 = 12.24, and 34.22 μM, respectively). Compound 3 showed strong cytotoxicity (GI50 = 4.4 μM) against MDA-MB-231 (an established basal triple-negative breast carcinoma (TNBC) cell line), with a high selective index of 35. This compound was subsequently evaluated for apoptosis induction in MDA-MB-231 cells, using GI50 and 50% lethal concentrations (LC50). Flow cytometry analysis showed that at LC50 compound 3 induced cell death with phosphatidylserine externalization and caspase-3 activation. Apoptotic genes were measured by RT-qPCR, revealing an upregulation of BAX, with an increase in expression of the BAX/BCL2 ratio in treated cells. Fluorescence microscopy disclosed morphological changes related to apoptosis. Overall, these findings showed compound 3 to be a promising prototype against TNBC cells that tend to respond poorly to conventional therapies.