Wall's heroics came after Al Horford's jumper with 7.7 seconds left put the top-seeded Celtics ahead 91-89.
Washington's Bradley Beal had 33 points and Wall scored 26 despite shooting 9 of 25 from the field.
Washington, which had lost seven straight elimination games at home, prevailed despite shooting 5 of 24 on 3-pointers and 43 percent overall. Boston shot 40.5 percent from the field.
Including the regular season, the teams are 10-0 at home in games against each other. Boston hosts Game 7 on Monday night. The winner will tip off the Eastern Conference finals Wednesday in Cleveland.
Thomas scored five straight points to put Boston up 87-82 with 1:34 remaining. Washington countered thanks to its dynamic backcourt. Beal, after missing all seven of his 3-point attempts, drained a shot from beyond the arc and Wall tied the game by hitting two free throws with 41.2 seconds remaining.
Lopsided results dominated the first postseason series between Boston and Washington since 1984 with each game decided by at least 10 points. After the Wizards won Games 3 and 4 by 27 and 19 points, respectively, the Celtics roared back with a 123-101 victory in Boston on Wednesday.
In Game 6, the Wizards found some life in a choppy first half, turning a 28-28 tie into a 40-30 lead on Jason Smith's free throw with 3:46 remaining.
Following 3-pointers from Thomas and Horford, Bradley's alley-oop dunk in the closing seconds sent Boston into halftime leading 42-41 despite shooting 35 percent from the field.
Wall continued his trend of slow starts. After missing his first nine shot attempts in Game 4 and opening 1 of 7 in Wednesday's road loss, the four-time All-Star missed 11 of 12 field goals Friday before hitting three in a row in the third. He scored 13 in the period as Washington entered the fourth trailing 69-66.
NOTES: Game 7 is slated to start at 8 p.m. Eastern time Monday night. ... Washington is 6-0 at home during the playoffs and 5-0 against Boston this season. ... Boston F Jae Crowder had 10 points. ... Washington's last victory in Boston came on April 16, 2014. ... The Celtics lost three of four in the head-to-head series with the Cavaliers during the regular season. Washington dropped two of three, but won in Cleveland 127-115 on March 25.