Upon its release, Hawk received positive critical acclaim. At Metacritic, which assigns a normalised rating out of 100 to reviews from mainstream critics, the album received an average score of 75, based on 22 reviews, indicating "generally favorable reviews."Allmusic's Mark Deming awarded Hawk four out of five stars and added that Campbell and Lanegan "continue to merge their distinct but complimentary styles while adding a few new edges to their approach."BBC Music reviewer Andy Fyfe referred to the duo as "a union that just keeps on giving" and called Hawk "teelier, more focused" and "its best album yet." In his review for the Chicago Tribune, Greg Kot gave the album three out of four stars and said the album had "a rich, insinuating sound", adding "there are also several twists that make this much more than just a rehash of a proven formula." Writing for Drowned in Sound, Alexander Tudor gave a mixed review, calling the album "the most energetic record yet from the pair" but adding that "some old weaknesses remain. Individually, and in collaboration, Campbell & Lanegan have always been limited to predictable melodies and chord progressions." Tudor rated the album six out of ten. Maddy Costa of The Guardian said "the pleasure of Campbell flitting like a will-o'-the-wisp in the cracks between Lanegan's fierce, parched growls is predictable now. Hawk impresses instead with its signs of Campbell's increased confidence as songwriter, arranger and producer", awarding the album three out of five stars.
Los Angeles Times reviewer Mikael Wood wrote a mixed review, awarding the album half of a four star rating, and stating the duo are "still drawing power from the tension between her wispy indie-pop coo and his deep blues-grunge growl" but "Hawk works best when Campbell [...] nudges their collaboration into new territory: the lush, retro-'70s soul sound of "Come Undone" [...] or "Get Behind Me," a blast of raw, White Stripes-style garage rock."Mojo rated the album four out of five stars and said that the duo had "surpassed the Nancy Sinatra/Lee Hazlewood comparisons to create an intense, fluid sound that's uniquely their own." Ash Dosanjh of the NME awarded the album a seven out of ten rating, praising Campbell's songwriting and vocals but adding "what is unfortunate is that she allows Lanegan to utterly dominate their duets." Writing for Slant Magazine, Jesse Cataldo said "their habit of slipping into archetypal styles often makes them seem shiftless and phony" and it "often sounds more like a jukebox than an album" but added it is "done so professionally, with a crisp production and a serious devotion to the particulars of each style." In his review for Spin, Marc Hogan rated the album six out of ten and said "Hawk faithfully follows its predecessors' dusty Americana blueprint."