"I think a lot of it is just each night I'm out there I've got run support," Moore said. "So, it makes it that much easier to pitch."
Moore (4-0) allowed one run, two hits, three walks and had nine strikeouts over eight innings. Roberts connected in the first and third to help the Rays take an early 5-0 lead.
"The thing that I saw tonight was that he has built off of the first three," Tampa Bay manager Joe Maddon said of Moore. "The first couple starts, he got through some moments, was not really sharp. But today he was sharp. Tonight was the first night I thought he put everything together since the beginning of the season. Matt was really the story tonight."
Over the past three games, Tampa Bay has held opponents to two runs -- including the final two games of a three-game sweep of Oakland. It was the third straight game Tampa Bay pitching has given up three or fewer hits.
"A lot of it is just feeding off each other," Moore said.
Tampa Bay has won 12 of its last 14 home games against the Yankees dating back to July 2011. Kyle Farnsworth completed the two-hitter.
Yankees infielder Kevin Youkilis, who left Saturday's game at Toronto with lower back stiffness, was a late scratch Monday after the back tightened up following batting practice.
"It might be a couple more days," Yankees manager Joe Girardi said. "Well see how he is tomorrow."
Sabathia (3-2) had a three-start winning streak end. The left-hander gave up five runs and seven hits, including three homers, over seven innings.
"I was trying to be more fluid with my delivery, and I think that took some of my aggressiveness away," Sabathia said. "I just have to be more aggressive, let the ball go and not think so much. I'm usually aggressive. It's just one of those things tonight where I don't know what I was thinking. I felt like I had better stuff than I've had, but it was a bad day."
The Yankees ace dropped to 10-10, including 3-6 in road decisions, against the Rays.
The Rays took a 4-0 lead in the first. After Roberts hit a solo homer on an 0-2 pitch from Sabathia, Evan Longoria singled and scored on Sean Rodriguez's triple. Yunel Escobar then put Tampa Bay up 4-0 with a two-run shot.
Roberts homered again in the third to make it 5-0. It was his third career multihomer game.
Sabathia had given up just one homer over 28 innings in his four previous starts this season. The Rays are the only team -- with three -- to have more than one game with three or more homers against him.
"You wouldn't necessarily think that any team would give C.C. a lot of trouble because he's been such a good pitcher, but they seem to give him more trouble here than at our place, and I'm not really sure why," Girardi said.
Robinson Cano cut the deficit to 5-1 on a homer in the fourth inning, which was New York's first hit. Cano also had the other hit off Moore, an infield single during the sixth.
The Yankees have homered in all nine of their road games this year, matching the team's longest streak since 2000.
The second game of the series Tuesday night has AL CY Young Award winner David Price (0-1) facing New York's Phil Hughes (0-2). The Rays have lost Price's four starts this season, which is the fifth time, and first since Ton Glavine in 1999, that a reigning Cy Young Award winner's team has lost his first four starts.
Star NFL CB Darrelle Revis, acquired by the Tampa Bay Buccaneers on Sunday from the New York Jets, threw the ceremonial first pitch. He received a big ovation from the crowd. ... Yankees OF Curtis Granderson (broken right forearm) hit off a tee and soft toss throws for the first time at the team's complex in nearby Tampa. ... Rays DH Luke Scott (strained right calf) is expected to start a minor league rehab assignment Tuesday at Class-A Charlotte. ... Sabathia has been scored upon in the first inning in two consecutive starts after a stretch of 13 starts where he kept the opponent scoreless in the opening inning. ... Longoria has reached base safely in 33 straight games dating back to last season.