The visiting Royals can record their highest win total in 20 years Tuesday night when they send Bruce Chen to the mound to face the Seattle Mariners.
Kansas City (83-73) will finish with a winning record for the first time in 10 years and can win 84 games for the first time since 1993.
Making the playoffs, however, will be difficult since the Royals trail Cleveland by three games for the AL's second wild card and are also two games behind Texas with six left for every club.
"Every game from now to the last game of the season is important to us," Royals catcher Salvador Perez said. "We can't think about the other teams, what happened. We just need a win. We can't control other teams. The only thing we can control is our game."
Dramatic wins to close a six-game homestand and open a seven-game trip, however, are cause for optimism. Justin Maxwell's grand slam in the 10th inning was the difference in Sunday's 4-0 win over the Rangers.
The Royals followed that up by blowing a two-run lead in the eighth inning Monday to open this three-game set before heroics by left fielder Alex Gordon and Perez in extra innings helped them avoid a crushing defeat.
That duo combined on the game's two key plays. Gordon threw out Kyle Seager at the plate with Perez making the tag for a double play that ended the 10th inning and scored on Perez's two-out double in the 12th to lift the Royals to a 6-5 victory.
"Alex Gordon's golden arm, and great tag play by Salvy," Royals manager Ned Yost said.
Closer Greg Holland walked the first two hitters in the bottom of the 12th before retiring the next three for his 45th save, tying the club record set by Dan Quisenberry in 1983 and matched by Jeff Montgomery 10 years later.
Kansas City is 11-4 in extra innings while Seattle has lost eight straight such games to fall to a major league-worst 6-15.
Six relievers combined to work the final 6 1/3 innings for the Royals, who will turn to Chen (8-3, 3.13 ERA).
The left-hander is 3-1 with a 3.86 ERA over his last five starts, allowing two runs in five innings last Wednesday to earn a 7-2 victory over Cleveland. Yost felt the veteran gives his team a boost in this race.
"I felt really good about Bruce being on the hill for us," Yost said.
Chen is 4-1 with a 2.63 ERA in 10 starts against Seattle, although Seager is 4 for 8 with two homers against him.
He has fared better versus Raul Ibanez (3 for 15), Justin Smoak (2 for 11) and Michael Saunders (3 for 12).
Seattle (68-89) is 21-29 when facing left-handed starters and will counter with fellow southpaw James Paxton (2-0, 2.12), who will make his fourth big league start.
Paxton, a 2010 Mariners fourth-round draft pick, won his first two outings before he gave up three runs in five innings Thursday in a 5-4 loss at Detroit.
Perez finished with his third three-hit game this season against the Mariners. He's batting .373 with 15 RBIs over his last 18 overall.
Seager is 14 for 33 (.424) during an eight-game hit streak versus Kansas City for Seattle, which has dropped five straight at home and 12 of 15 overall.