The Rays have won 20 of 23 and took sole possession of the AL East lead for the first time since June 10, 2012. Tampa Bay moved a half-game ahead of Boston.
Ben Zobrist, Yunel Escobar and Evan Longoria doubled during a six-run burst in the second inning, and James Loney later homered and drove in four runs.
Well supported all season, Hellickson (10-3) won his sixth straight decision. Fernando Rodney, the Rays' fourth pitcher in a three-run ninth, got the last out for his 25th save.
Trying to add power, the Yankees acquired Soriano from the Chicago Cubs before the game, trading a minor league pitcher to get the seven-time All-Star and lots of cash to cover much of his contract.
Soriano got his chance to make a sudden impact, batting with the bases loaded, two outs in the third and the Yankees trailing 6-1. With Tampa Bay's infield overshifted to the left side, he ran the count full before lofting a routine fly to right that ended the inning.
The 37-year-old Soriano, who began his career with the Yankees in 1999 and drew a big ovation in his return, went 0 for 5, scored once and drove in a run. He got an RBI with a bases-loaded forceout in the ninth.
Soriano did more on defense, catching a fly to start the game, making a throw that helped nail a runner and running down a foul ball.
Hellickson gave up one run and four hits in six innings, retiring his last nine batters and sending the Yankees to their seventh loss in 10 games. He also flashed a fast glove, snaring Ichiro Suzuki's liner right in front of his face.
Sabathia (9-9) was booed early and was hit hard for the third straight start, allowing 22 runs over 14 innings in that span. He got a true Bronx cheer when he struck out Loney while escaping a bases-loaded, no-out jam in the third.
The Rays have been particularly prickly for Sabathia. He is just 4-11 in 23 starts against them, including four this season, since joining the Yankees.
A two-out walk to eighth-place hitter Jose Lobaton set up Sam Fuld's RBI single in the second and Desmond Jennings followed with a bunt single that catcher Austin Romine threw away for another run. Longoria delivered a two-run double and scored on Wil Myers' single.
Loney had an RBI single in the fifth and hit a three-run homer in the seventh off Adam Warren that made it 10-1.
NOTES: Rays LHP David Price will start Monday's makeup game at Boston. He won Wednesday night at Fenway Park, pitching a five-hitter. ... Rays rookie RHP Chris Archer (5-3, 2.76) starts Saturday vs. RHP Ivan Nova (4-3, 3.66). ... Rays RHP Alex Cobb has a bullpen session Saturday in Port Charlotte, Fla. Hit in the head by a line drive last month, he threw two pitches in his first rehab start before being pulled because of a blister. ... Yankees star Curtis Granderson (broken left pinkie) played LF and went 0 for 3 with Class A Tampa. 3B Jayson Nix (strained right hamstring) hit an RBI double in four at-bats. Rays RHP Brandon Gomes (strained right lat) pitched 1 2-3 scoreless innings in the game. ... The Yankees have signed minor league INF Brendan Harris.