Positron emission tomography (PET) imaging is an invaluable tool in clinical settings as it captures the functional activity of both healthy anatomy and cancerous lesions. Developing automatic lesion detection methods for PET images is crucial since manual lesion segmentation is laborious and prone to inter- and intra-observer variability. We propose a 3D disentanglement method that learns robust disease features and predicts lesion segmentations by disentangling PET images into disease and normal healthy anatomical features. The proposed method, PET-Disentangler, uses a 3D UNet-like encoder-decoder architecture for feature disentanglement followed by simultaneous segmentation and image reconstruction. A critic network encourages the healthy latent features, which are disentangled from disease samples, to match the distribution of healthy samples and thus do not contain any lesion-related features. We train and evaluate PET-Disentangler on 3D PET images from the Cancer Imaging Archive (TCIA) whole-body FDG-PET/CT Dataset consisting of 1014 PET/CT scans, leveraging TotalSegmentator to obtain two anatomically aligned field-of views of the whole-body scans referred to as the upper and lower torso regions. Compared to non-disentanglement segmentation methods, our quantitative results on the upper torso region show PET-Disentangler has similar performance while having the added advantage of visualizing, via the pseudo-healthy image, how a healthy (lesion-free) image might look like. Our quantitative and qualitative results on the lower torso show enhanced performance from our method as PET-Disentangler reduces the chances of incorrectly declaring high tracer uptake regions as cancerous lesions, since such uptake pattern would be assigned to the disentangled normal component.