The exquisite angular resolution and sensitivity of JWST are opening a new window for our understanding of the Universe. In nearby galaxies, JWST observations are revolutionizing our understanding of the first phases of star formation and the dusty interstellar medium. Nineteen local galaxies spanning a range of properties and morphologies across the star-forming main sequence have been observed as part of the PHANGS-JWST Cycle 1 Treasury program at spatial scales of ∼5–50 pc. Here, we describe pjpipe, an image-processing pipeline developed for the PHANGS-JWST program that wraps around and extends the official JWST pipeline. We release this pipeline to the community as it contains a number of tools generally useful for JWST NIRCam and MIRI observations. Particularly for extended sources, pjpipe products provide significant improvements over mosaics from the MAST archive in terms of removing instrumental noise in NIRCam data, background flux matching, and calibration of relative and absolute astrometry. We show that slightly smoothing F2100W MIRI data to 0.″9 (degrading the resolution by about 30%) reduces the noise by a factor of ≈3. We also present the first public release (DR1.1.0) of the pjpipe processed eight-band 2–21 μm imaging for all 19 galaxies in the PHANGS-JWST Cycle 1 Treasury program. An additional 55 galaxies will soon follow from a new PHANGS-JWST Cycle 2 Treasury program.