The 48 points matched a franchise high set 14 years ago against the Atlanta Falcons.
Tampa Bay's Ronde Barber had three interceptions to set a team single-game record, including one he returned 36 yards for a touchdown.
"It was real fun. I was in position to make a couple of plays," said Barber, who is tied with Cleveland Browns rookie Anthony Henry for the league lead in interceptions with nine. "It was an overall effort and we were able to create some turnovers and it was our type of game."
Sunday's game, which matched teams that will move to the NFC South next season, will likely go a long way to determining the third and final NFC wild card berth.
The win was the fourth in five games for the Buccaneers (8-6), who moved a game ahead of New Orleans (7-7).
The Saints were coming off a Monday night loss to the St. Louis Rams and did not appear ready to play.
"We were bad all around in all three phases, but we have to give credit where credit is due," Jim Haslett said. "We didn't run well and we didn't tackle well. They beat us in every phase today. They scored right away, we threw an interception and we never had a chance."
"I think we took advantage of a Monday night hangover," Buccaneers coach Tony Dungy said referring to the Saints' loss to St. Louis. "For us, it was a good game and it came at a good time."
On the other hand, the Buccaneers displayed plenty of enthusiasm after they were embarrassed at Chicago last week. Tampa Bay crushed the Saints in the first half, totaling 269 yards and 15 first downs to just 34 yards and one first down for New Orleans.
Johnson threw for 165 yards and two scores in the half while New Orleans quarterback Aaron Brooks passed for just 33 yards.
The Buccaneers rushed for 104 yards in the half, with Alstott gaining 69 on 14 carries. Using his 248-pound frame to bounce off tacklers, Alstott bulled through linebacker Kevin Mitchell on his one-yard touchdown run in the first quarter.
"It was just an all-around great day," Alstott said. "The guys up front did a heck of a job against a very, very good defensive line and they did not give up a sack."
Alstott finished with 101 yards for his second 100-yard game of the season.
New Orleans' Ricky Williams was a non-factor. He rushed for only seven yards on five carries in the first half.
The lone negative for Tampa Bay was an injury to kicker Martin Gramatica, who converted four field goals but strained his right hamstring kicking an extra point in the fourth quarter.
"We're not sure if it is tweaked or a cramp," Gramatica said. "I felt it the most on a kickoff. There was no sense going back in and injuring it. It feels fine now and I will be ready for Saturday (against Baltimore)."
Tampa Bay has the advantage of playing its final two games at home, but must contend with the defending Super Bowl champion Baltimore Ravens and Philadelphia Eagles.
"We need to crank the power and mash the gas," said Bucs defensive tackle Warren Sapp, who played with a bruised shoulder. "We are in control of our destiny now."
This was the final road game for the Saints, who close the season with home games against Washington and San Francisco.
"It's not over yet," said Saints receiver Joe Horn, who caught two touchdown passes. "We have to hope that Tampa Bay slips a notch. I'm not going to hang my head right now because if and when we get into the playoffs, everyone is back to 0-0."
Stecker set the tone for the game when he returned the opening kickoff 86 yards -- the longest in club history -- to the New Orleans 14. Stecker broke numerous tackles on the return before he was taken down by Michael Lewis.
"It wasn't necessarily the return itself, it was the attitude during the return," Bucs linebacker Derrick Brooks said. "He could have had an 86-yard return without getting touched, but it was Aaron breaking tackles, it was guys finishing their blocks. It was just the whole nature of the return, to me, was an attitude."
Two plays later, Karl Williams caught a short pass from Johnson, eluded a tackle by cornerback Kevin Mathis and turned it into a 14-yard touchdown 1:06 into the game.
"You talk a lot in games like this about tempo setters. Aaron Stecker did a real good job on the kickoff return," Bucs safety John Lynch said. "He let the Saints know we were coming to play today. Plays like that lift the whole team. It lifted the electricity in the stadium and we kept it going on defense."