The Rays' right-hander continued his comeback from Tommy John surgery that sidelined him most of the past two seasons, pitching into the sixth inning of a 4-1 victory over the New York Yankees on Wednesday night.
Building on five starts he made late last year, Cobb (1-0) allowed one run and four hits in 5 2/3 innings. He walked one and struck out four.
"That was probably the best I've been on the mound since I've been back from Tommy John," Cobb said. "It's not to the point where I would like to end up being eventually, but it's definitely good enough to go out there and compete with."
Dickerson hit the first leadoff homer of his career , driving Michael Pineda's third pitch into the right-field seats. He added an RBI single in the second, when Tampa Bay scored three times with two outs. Norris drove in the first two runs of the inning with a single to center off Pineda.
"It just came down to he made some mistakes with his fastball," Yankees manager Joe Girardi said. "He made one to Dickerson early. The big one he made was to Norris. He had two outs and he looked like was going to get out of that inning without giving up a run. Just couldn't seem to get the third out."
Jacoby Ellsbury had three hits, including a home run off Cobb , but the Yankees got little production from the heart of their lineup in the series. Chase Headley had his third straight multi-hit game for New York, going 2 for 3 to finish 7 of 11 in the three games.
Gary Sanchez, Greg Bird and Matt Holliday -- the Nos. 2, 3 and 4 hitters -- went 4 for 37 with one RBI in the series. Sanchez, off to a 1-for-14 start, singled off Cobb in the third inning for his first hit.
"They got off on a little slow start," Girardi said, "but I'm not worried."
Pineda (0-1), who was 0-3 with a 7.30 ERA in five starts against Tampa Bay a year ago, struck out six while allowing four runs and eight hits over 3 2/3 innings.
If there was a bright spot for the Yankees in the series, it was the performance of a bullpen that didn't allow a run in 13 2/3 innings, including the final 4 1/3 innings Wednesday night.
Cobb, meanwhile, made his first April start in three years. He won at Tropicana Field for the first time since Aug. 21 against Detroit.
The right-hander missed the entire 2015 season after undergoing elbow surgery. He went 1-2 with an 8.59 ERA last season.
"This entire time I've tried not to be results-based, whether I'm happy or not with myself with the way the comeback's coming," Cobb said. "There's definitely some more things on my end I need to get to, but with the stuff I had today, I'm real happy about it."
Yankees: SS Didi Gregorius, out until sometime in May with a sprained right shoulder, will increase his throwing from 60 to 90 feet on Thursday.
Rays: SS Matt Duffy (Achilles tendon), one of seven Tampa Bay players who began the season on the disabled list, did flexibility drills in the outfield.
Headley, who got off to a terrible start last year, has three consecutive multi-hit games to begin a season for the first time. It's the longest such streak for the Yankees since Alfonso Soriano had multiple hits in the first six games of 2003. Headley had a double and a single Wednesday night after going 5 for 8 in the first two games, when he singled three times through vacated spaces on the left side of a shifted infield.
Rays manager Kevin Cash said Headley's success against the shift won't necessarily force the Rays to re-think their defensive strategy moving forward.
"I might be wrong, but we're going to trust our process in how we position guys. And if we circle back over a month or a couple series time, and it's shown that enough throughout the league, then we'll make an adjustment," Cash said. "But I don't think we can make a quick adjustment after two games."
Yankees: Following an off day, RHP Luis Severino faces Orioles RHP Ubaldo Jimenez on Friday night in the first of three games in Baltimore. Severino won the fourth spot in the rotation during spring training. He went 0-8 with an 8.50 ERA in 11 starts last season.
Rays: LHP Blake Snell and Toronto RHP Marcus Stroman are the scheduled starters Thursday in the opener of a four-game series. Stroman is 3-1 with a 2.89 ERA in four career starts at Tropicana Field.
More AP baseball: apnews.com/tag/MLBbaseball