Catchings had 13 rebounds while Marissa Coleman added 11 points for Fever (1-3), who got their first win under head coach Stephanie White.
Catchings made a layup with 1:23 to play and, after grabbing a rebound on the other end, found Natalie Achonwa for a layup to make it 73-69 with 58 seconds left.
Leilani Mitchell hit a 3-pointer to pull Phoenix (2-1) within one with 8 seconds left.
Catchings' free throws made it 77-74 a second later and Mitchell was off the mark on a deep 3 as time expired that could have tied the game.
Mitchell scored a career-high 25 points and was 7 of 10 from 3-point range for the Mercury, while DeWanna Bonner, who came into the game averaging 21.5 points, had 12 on 4-of-19 shooting.