''He believes he belongs in the big leagues and we've felt that way all along,'' Tampa Bay manager Joe Maddon said. ''You can only have so many starting pitchers.''
Cobb (1-0) allowed two runs and six hits in seven innings. He retired his last 10 batters.
Cobb was recalled from Triple-A Durham on Friday night to take the place of Jeff Niemann, who is on the 60-day disabled list with an injured right leg. The 24-year-old righty went 3-2 in nine starts for Tampa Bay last season.
''Cobb was phenomenal,'' Joyce said. ''In the first inning or two, he had the adrenalin and he was excited, but he settled in really well. He's going to be a big part of our success.''
After Tampa Bay loaded the bases on a pair of a walks and a single in the third, Joyce put the Rays up 4-2 when he sent a 2-0 pitch from Randall Delgado (2-4) over the center-field fence.
''Usually when a pitcher falls behind 2-0, they're going to try to throw a strike and that's exactly what I was looking for,'' Joyce said.
The Braves had scoring chances in each of the first four innings, but Cobb induced two double-play grounders.
''My game is to get groundballs, and they started finding a few holes,'' Cobb said. ''To be able to get a little momentum going on the double plays was huge.''
Atlanta went 1 for 5 with runners in scoring position.
''We kind of let him off the hook I think there early,'' Braves catcher Brian McCann said. ''Then he fell into a rhythm and he got ahead of everybody after that. The fourth inning, he started really pounding the zone, really trusting his fastball.''
Braves third baseman Chipper Jones was out of the lineup because of a bruised left calf. He took a hard one-hop shot off the bat of B.J. Upton in Friday night's 5-3 win over the Rays. Atlanta manager Fredi Gonzalez is hopeful that Jones will be able to play again by Monday.
Sean Rodriguez extended the Rays' advantage to 5-2 on a sixth-inning solo homer.
''You felt like you were in the game the whole time, you really did,'' Gonzalez said.
Delgado gave up four runs, three hits and walked five in four innings.
''It was a learning start for him,'' Gonzalez said. ''Just fell behind hitters. When you're dealing with a young kid, there's going to be times like this.''
Rhymes returned to lineup after missing two games with a bruised right forearm. He was hit by a pitch on Wednesday, fainted after walking to first base and had to be driven off the field on a cart.
NOTES: Joyce has three career grand slams. ... Tampa Bay 3B Evan Longoria (partial tear left hamstring) increased his agility work on the infield while fielding grounders before the game. ... Atlanta RHP Tim Hudson (2-1) and Tampa Bay LHP David Price (6-2) are Sunday's scheduled starters. Hudson has a 7-1 career mark against the Rays. ... Tampa Bay RHP Kyle Farnsworth (strained right elbow) is throwing at 60 feet on level ground and long tossing.