Rookie Drew Smyly, who had started warming up in the third when starter Doug Fister took a line drive off his right wrist, got the win by pitching two scoreless innings, ending a 4-hour, 54-minute marathon.
In Game 2 on Sunday, New York will start Hiroki Kuroda, who will be pitching on three days' rest for the first time in his big league career. Detroit will send Anibal Sanchez to the mound.
Twenty-five of 42 previous Game 1 winners have gone on to take the AL pennant.
Before the 12th, the star of the night was Ibanez, the first player to hit three home runs in the ninth inning or later in a single postseason.
On Wednesday, he hit a tying shot as a pinch hitter, and three innings later became the first player to hit two homers in a postseason game he didn't start.
This made him the first player in baseball history with two tying ninth-inning home runs in a single postseason, according to STATS LLC. Cincinnati's Johnny Bench, in 1972 and 1976, had been the only player to do it twice in a career.
Fister escaped three bases-loaded jams in the first six innings — the first time in their 375 postseason games the Yankees stranded a trio of runners three times without scoring in any of those innings, according to STATS LLC.
Alex Rodriguez, back in New York's lineup following a benching in Friday's division series finale, was 0 for 3 and stranded six runners — striking out on three pitches with runners at second and third and no outs in the sixth as fans booed loudly.
Girardi sent up Eric Chavez to hit for A-Rod in the eighth, the third time Rodriguez was taken out early in the last three games he has played.
Not that A-Rod's teammates were any better. The Yankees stranded 13 runners and were 3 for 13 with runners in scoring position, leaving them at 10 for 45 (.222) in the playoffs.
Fister, who beat the Yankees in Game 5 of last year's division series, had a shaky start, walking the bases loaded in the first and allowing three two-out singles in the second.
Rodriguez bounced into a forceout that ended the first, with shortstop Peralta making a spectacular diving stop. In the second, Cano lined a ball off the inside of Fister's right wrist, and Peralta picked up the ball on a bounce and threw to first for the out. Both plays were so close that even replays didn't definitively show whether the calls were correct.
Detroit was so concerned about Fister's wrist that Smyly started warming up in the third. Fister changed from a short sleeve undershirt into long sleeves and stayed in the game for 6 1-3 scoreless innings. His finest moment came with Detroit leading 2-0 in the sixth. After fanning Rodriguez, he loaded the bases with a walk to Swisher, then struck out Curtis Granderson and Martin with breaking balls.
Fister allowed six hits, struck out five and walked four.
"It was a little stiff, little sore, but nothing too major," he said of his wrist. "I went in and checked it out, and made sure nothing was remarkably hurt. Put on some sleeves to keep warm, and went back out there."
Rodriguez was dropped to sixth in the batting order for the first time since the 2006 playoff finale against Detroit, but the key situations find the $275 million man no matter where he is. He is hitting .105 (2 for 19) with no RBIs in the postseason, going hitless in 15 at-bats against right-handed pitchers with 10 strikeouts. A-Rod hasn't homered in 87 at-bats since Sept. 14.
Cano (2 for 28), Granderson (3 for 23) and Swisher (3 for 23) also remained mired in deep postseason slumps, with Swisher's eighth-inning double the only hit among the three.
Making his record 44th postseason start, Pettitte gave up two runs and seven hits in 6 2-3 innings.
Afterward, the Yankees clubhouse was as quiet as it ever is. The loss of Jeter weighed on the minds of his teammates more than the defeat,
"I think we probably feel more for him than anyone else because we know how important it means to him personally," Teixeira said.
NOTES: Leyland is 2-4 in previous LCS he has managed. ... On a cool, 49-degree night, there were empty seats in the top two decks in the outfield for the second straight day.