Those homers accounted for all four runs off Josh Lindblom (0-2), who went six innings on his 26th birthday. The right-hander struck out two in his third start in place of injured Alexi Ogando.
Dickey (6-8) struck out three against the team that took him in the first round of the 1996 draft, and later determined that the right-hander was missing a ligament in his right elbow.
"It's somewhat nostalgic to come back here," said Dickey, who also won his only other start at Rangers Ballpark as an opponent, in 2008 with Seattle. "This is where the discovery about the absence of the ligament was made, this is where I started my knuckleball career. I have a lot of great memories here, and a lot of tough memories here."
Texas, which lost 8-0 on Friday night, ended a 21-inning scoring drought in the sixth. Nelson Cruz and David Murphy had consecutive bloop singles before Chris McGuiness grounded a hit through the right side of the infield. Cruz scored when Leonys Martin beat out a potential inning-ending double-play grounder.
Martin was 3 for 4, getting the RBI on the only at-bat he didn't get a hit.
His fielder's choice ended Dickey's day and the longest scoreless streak by the Rangers since 25 innings in a row without a run in September 2009, also the last time they lost five games in a row at home.
Texas also had the bases loaded in the fourth before McGuiness struck out to end the inning. There were two runners on with no outs in the fifth before Dickey retired the Nos. 2-4 hitters in order.
"We got (Dickey) in trouble a few times. He got out of it," Washington said. "Some of it was him, but most of it was us."
Kinsler, out since May 18 with a ribcage injury, was supposed to stay at Double-A Frisco for at least three games and not expected back before Sunday at the earliest. That changed when after an 11-inning game on Friday night, he declared himself healthy and ready to rejoin the team — even after going 0 for 8 with a walk in the two rehab games.