"You always want to avoid a sweep and finish a series well," Sabathia said. "It felt good to get a win."
Victories have been rare so far this year for the banged-up Yankees.
New York (2-4) needed to beat the Tigers to avoid dropping to its worst six-game record since starting the 1989 season 1-7.
"The only must-win you need is when you're about to be eliminated," Yankees manager Joe Girardi said.
Sabathia (1-1) gave up four hits and three walks while striking out four in 114 pitches, bouncing back from a dreadful Opening Day start.
"He's one of the better pitchers in baseball and he showed that," Tigers manager Jim Leyland said.
The Yankees don't have their best players in the lineup because All-Stars Derek Jeter, Alex Rodriguez, Mark Teixeira and Curtis Granderson are on the disabled list, but Leyland said it would be foolish to count them out this season.
"They're missing some pieces, but they're always going to be good," Leyland said. "They'll be right in the thick of things like they always are."
The Yankees got all the run support they needed in the second inning when Francisco Cervelli hit an RBI double and Nix followed with a two-run, two-out homer to give the Yankees a 3-0 lead.
Verlander (1-1) allowed three runs on seven hits and three walks. Verlander said he didn't have a bad game, or inning, but lamented the high, hanging changeup he threw to Nix that ended up over the left field fence.
"Poor pitch selection and poor pitch execution on the same pitch usually doesn't end up too well," he said.
Yankees reliever David Robertson entered the game in the eighth and got out of a two-on, one-out jam by getting Victor Martinez to hit a fly to shallow center and Matt Tuiasosopo to strike out. Mariano Rivera pitched the ninth, drawing some cheers after he was honored in a pregame ceremony for his planned retirement. Rivera gave up two hits and ended the game by striking out Torii Hunter.
The Tigers gave Rivera a plaque with pictures of him pitching in Detroit along with dirt from Comerica Park and Tiger Stadium.
"That was pretty neat," Rivera said.
After losing the first two games in Detroit by a combined score of 16-7, the Yankees put a strong performance together at the plate on the mound and in the field.
"Pitching dictates everything, and CC pretty much dictated the pace," Leyland said.
Detroit had won six straight against the Yankees, including sweeping them in last year's AL Championship Series. The Tigers advanced to the World Series by roughing up Sabathia in Game 4, scoring six runs -- five earned -- and getting 11 hits off him in just 3 2/3 innings. His first start this year wasn't much better.
Sabathia gave up four runs and eight hits over five innings in a season-opening, 8-2 loss to Boston, and learned from the loss.
"I pitched in a lot more and a lot more effectively," he said.
Yankees SS Eduardo Nunez wasn't in the lineup for second straight day after getting hit on his right biceps, but entered the game as a pinch-runner in the eighth inning. ... The Tigers are off Monday before opening a home series against Toronto on Tuesday afternoon. ... Yankees RHP Hiroki Kuroda, who bruised his right middle finger in his last start, is scheduled to pitch Monday at Cleveland. ... The Tigers drew 127,333 fans to three-game series, the largest opening series crowd in Detroit since 1948. ... New York OF Brennan Boesch received his AL championship ring in the clubhouse Sunday morning. ... Hunter has five multihit games in his first six with the Tigers. ... Nix hit his first homer since July 28 of last year. ... Yankees RHP Joba Chamberlain shaved his moustache, saying it wasn't doing the team any good.