Catchings' 3-pointer with 1:34 remaining in overtime put the Fever ahead for good, 75-74. With that basket, she surpassed Lisa Leslie as the WNBA's career leading scorer in the postseason.
Kara Lawson scored a season-best 20 points for the Mystics, who led 62-57 with 3:57 remaining in regulation. Including the regular season, Washington lost five of its final six games.
"It's never fun to lose in a playoff series especially when you have a chance to win," Mystics coach Mike Thibault said. "I guess I can say we got beat by a great Hall of Famer. (Tamika) was terrific."
The Fever will face top-seeded Atlanta or No. 4 Chicago in the conference finals.
Briann January scored 13 for Indiana. Her two free throws with 19.7 second remaining in regulation tied it 66-66. Marissa Coleman and Karima Christmas both added 10 points.
The Fever offset poor shooting from the field by making 22 of 26 foul shots. Catchings, now with 514 career playoff points, sank 9 of 10 including four in the final 21 seconds after making all 14 of her attempts in Game 1.
Indiana held Washington's leading scorer Ivory Latta without a 3-point basket for the first time this season. Latta scored 12 points on 2 of 13 shooting, including seven attempts from beyond the arc.
"I thought everything we did tonight (defensively) was focused on Ivory Latta," said Indiana coach Lin Dunn, who previously announced she will retire following this season.
Indiana won all three games in Washington this season.
The Fever also swept the Mystics in 2009. Washington last won a playoff series in 2002.
Tianna Hawkins scored nine of her 13 points in the fourth quarter for the youthful Mystics.