The White Sox went 8-1 on their road trip and reached .500 for the first time since they were 1-1 in early April.
"It's been a long road to get back," Konerko said. "We dug ourselves a big hole. Now we can try and move from here, set another goal -- maybe pick up a game or two this week and just keep climbing."
Chicago once again stuck to what is becoming a familiar formula -- a strong outing from a starting pitcher and sufficient offense without the benefit of a home run.
Garcia (8-3) gave up three runs and six hits over seven innings. He struck out six, walked one and won for the eighth time in nine starts.
White Sox starters are 9-1 with a 1.97 ERA in their past 12 games, and the rotation's dominance has Chicago hitters feeling little pressure.
"Our pitching has been good," Konerko said. "We feel like if we can get them some runs early, we're in the driver's seat."
Perhaps so, but Sunday's win marked the eighth straight game in which the White Sox have not homered, their longest power outage since an eight-game drought from April 17-25, 1993.
But with a winning streak and an opportunity to get over .500 when they return to U.S. Cellular Field on Tuesday, the lack of home runs can be overlooked, said manager Ozzie Guillen.
"I thought the way we started the season, it would be very, very tough to climb this ladder and all of a sudden we're here," Guillen said.
The Nationals, who lost their season-worst sixth straight game, are headed in the opposite direction. It's Washington's longest skid since dropping eight in a row from Aug. 28-Sept. 5, 2009, and the Nationals are a season-worst eight games under .500.
"It's just not happening for us right now," said Washington manager Jim Riggleman. "We are struggling -- everybody in there knows it. We're sticking together, though."
Adam Dunn insisted the Nationals aren't experiencing the kind of poor play that resulted in successive 100-loss seasons in 2008 and 2009."
"It's just a little rut we're in," Dunn said. "We weren't very good last year. That's why we lost 100 games. We are a lot better than that."
Alex Rios added three hits and two RBIs for Chicago.
Down 3-1, the White Sox opened the fifth inning with six straight hits and chased John Lannan (2-6). Konerko put Chicago ahead with a two-run single.
Konerko gave Chicago a 1-0 lead in the first with a two-out single that drove in Rios, who singled and stole second.
Garcia didn't allow a hit through three innings, fanning five, then ran into trouble in the fourth when the Nationals went ahead 3-1.
The White Sox retook the lead in the fifth. After singles by Juan Pierre and Alexei Ramirez, Rios doubled to cut Washington's lead to 3-2. Konerko followed with a two-run single up the middle, Chicago's 11th hit.
Rios drove in a ninth-inning insurance run.
The interleague series drew 108,575, the largest three-game attendance at Nationals Park this season. ... Nationals CF Nyjer Morgan made a nice running grab in right-center to rob Viciedo of an extra-base hit in his first major league at-bat in the second. Viciedo singled to center in the fourth and finished 1 for 4 in his debut. ... Bernadina has hit safely in 14 of 17 June games. ... Castro was credited with his second career stolen base -- his first since 2005 -- in the second when SS Ian Desmond couldn't get C Nieves' pickoff throw out of his glove, allowing Castro to reach third. The play was originally scored as an error by SS Desmond, but changed before the fifth inning. ... Following the game, fathers and sons celebrated Father's Day by playing catch in the outfield at Nationals Park.