Giambi drove a 1-1 pitch from Addison Reed (5-4) into the lower deck in right field to save the Indians from a potentially devastating loss. Giambi's shot bailed out controversial closer Chris Perez, who gave up two homers in the top of the ninth.
As he rounded third base, the 42-year-old Giambi slowed down before being engulfed by his teammates as the Indians celebrated their 13th straight win over Chicago.
Bryan Shaw (6-3) came in after Perez had nearly given away a game the Indians couldn't afford to lose.
RAYS 7, YANKEES 0
NEW YORK (AP) — Matt Moore had New York flailing for five wild innings, and the Rays beat the Yankees to push them to the brink of missing out on the post-season for the second time in 19 years.
Tampa Bay roughed up Hiroki Kuroda in tightening its grip on an AL wild-card spot with its fifth straight win. Matt Joyce had a leadoff homer as the Rays jumped ahead 3-0 in the first and David DeJesus had two RBI doubles.
Tampa Bay has a one-game lead over Cleveland for the top wild card with five to play. Texas and Kansas City played later.
The Yankees trail the Indians, who won 5-4, by five games — and also are behind the Rangers and Royals. The Yankees would be eliminated Wednesday if Cleveland wins and they lose.
Moore (16-4) gave up only three hits but threw three wild pitches to up his AL-leading total to 16 and walked six.
TIGERS 4, TWINS 2
MINNEAPOLIS (AP) — Doug Fister struck out seven and the Tigers homered three times in the fourth inning to clinch a playoff berth with a victory over the Twins.
Victor Martinez and Omar Infante went back-to-back and Austin Jackson hit a two-run homer later in the fourth to put the Tigers in front. Fister (14-9) gave up two runs and eight hits in 6 1-3 innings to help Detroit secure at least an AL wild card.
Cleveland rallied to beat the Chicago White Sox 5-4 in the bottom of the ninth inning to prevent the Tigers from clinching their third straight AL Central title. One more Tigers win or Indians loss will lock it up for Detroit.
BLUE JAYS 3, ORIOLES 2, 10 innings
BALTIMORE (AP) — The Toronto Blue Jays eliminated Baltimore from the playoff hunt, using two clutch RBI singles by Mark DeRosa to beat the Orioles.
Baltimore's sixth straight loss, combined with Cleveland's comeback win over the Chicago White Sox, left the Orioles six games behind the Indians for the final wild-card slot with five games left.
DeRosa singled in the tying run in the eighth inning and put the Blue Jays ahead in the 10th.
Sergio Santos (1-1) worked a perfect ninth and Casey Janssen got three outs for his 34th save.
Jose Reyes drew a one-out walk in the Toronto 10th off Francisco Rodriguez (2-1) and Munenori Kawasaki singled. Reyes appeared to be picked off second but was called safe, drawing Orioles manager Buck Showalter out to argue the call. Brett Lawrie then hit into a fielder's choice and DeRosa singled to right.
RANGERS 3, ASTROS 2
ARLINGTON, Texas (AP) — Adrian Beltre homered at home for the first time in more than a month and the Texas Rangers stayed close in the wild-card race, beating Houston and sending the Astros to their 11th straight loss.
With five games left in the regular season, Texas remained a game behind Cleveland for the AL's second wild-card spot. Tampa Bay is a game ahead of the Indians in a crowded pack.
Beltre's 29th homer snapped a 2-all tie with one out in the sixth. That came after Yu Darvish gave up the lead in the top half of the inning.
Matt Dominguez homered for the Astros. Their 107th loss overall matched the franchise record set last year.
MARINERS 4, ROYALS 0
SEATTLE (AP) — Justin Smoak hit a three-run homer in the fifth inning and the Mariners beat Kansas City, putting the Royals' post-season hopes in serious jeopardy.
Mariners rookie left-hander James Paxton (3-0), making just his fourth major league start, worked a career-high seven innings, allowing four hits, walking none and striking out a career-high 10.
The playoff picture looks as bleak as it can be for the Royals (83-74). They are four games behind Cleveland for the second wild-card berth with five games to play.
The Royals' three competitors for the two wild-card spots — Tampa Bay, Texas and Cleveland — all won Tuesday. The Royals need to win every game and have both the Indians and Rangers completely collapse.
It was the 13th shutout for the Mariners and the ninth time the Royals have been shut out.
ANGELS 3, ATHLETICS 0
ANAHEIM, Calif. (AP) — Jason Vargas earned his second shutout of the season with a four-hitter and Howie Kendrick homered, leading the Angels to a victory over the Athletics.
The Athletics, who clinched their second straight division title Sunday, remain a game behind the Red Sox for the best record in the majors following Boston's 8-3 loss at Colorado.
Vargas (9-7) threw 110 pitches, retiring 15 consecutive batters at one point and striking out five en route to his fifth shutout and 10th complete game in 153 career starts. The left-hander, who was acquired from Seattle last December for slugger Kendrys Morales, is 3-3 with a 4.64 ERA in his last nine outings after missing almost two months because of a blood clot near his armpit.
Griffin (14-10) gave up three runs and five hits over five innings.