James scored 29 points to lead the Lakers, but they were unable to dig themselves out of a 29-point first-half hole and lost the best-of-seven Western Conference series four games to two.
The last defending NBA champion to be knocked out in the first round was the San Antonio Spurs in 2015.
"That's the only way we wanted it to be honest," Booker said of drawing the Lakers in the first round.
"We knew we weren't going to get to where we wanted to go without going through them, and it happened to be in the first round.
"It was a tough match-up all the way through. Once they got us down 2-1 we had to regroup and get everything right and we came in and we battled," added Booker, whose team next faces the Denver Nuggets, who eliminated the Portland Trail Blazers Thursday with a 126-115 victory in game six of their series.
James, who won his fourth NBA title when he led the Lakers to victory in the quarantine bubble in Orlando last season, had never before lost in the first round in 14 prior appearances.
But he and the Lakers clearly felt the lack of star Anthony Davis, who was a game-time decision to start after missing the second half of game four and all of game five with a groin strain.
Davis, clearly hindered by the injury, lasted less than six scoreless minutes and the Suns grabbed the game by the throat in the first quarter with Booker scoring 22 points in the opening period.
"He's a warrior," Vogel said of Davis. "It was difficult for him to be able to move. We did everything we could from a treatment standpoint to loosen it up and we were hoping that some in-game adrenaline would help.
"But he certainly wasn't moving well at all to start the game and then went down grabbing at it early in the first."
With the series in the balance, Davis said, "I obviously didn't want to just lay down.
"I wanted to do what I can," he said. "It never felt good, but the competitive nature in me wanted to go out and help the team as best as I could. But my body didn't agree."
Davis and James both said that a series of injuries along with absences due to the league's Covid-19 protocols prevented the champions from fulfilling their potential.
"Obviously (we) fell short of our goal, but we tried to do everything we could to be as good as we could be," James said.
Meanwhile, Booker made eight of nine shots from the floor in the first quarter, including six of six from three point range.
The Suns led 36-14 after the first period and from there the Lakers never got the deficit below double digits.
"Credit the Suns," Vogel said. "They came out and made their first seven threes or something. Booker was red-hot to start the game. They put us in the deficit and it was too much to overcome."
The Suns have won three straight games to reach the second round for the first time since 2010, when they fell to the Lakers in the Western Conference finals in their most recent playoff appearance.
"I've been working my whole life for this moment so it wasn't time to shy away from it," Booker said. "Plenty of hours of sacrificing a lot of life to this game. So I wanted to put it on full display tonight."
But he said the Suns would face another demanding test in the Nuggets.
"It's going to be a battle and we understand that going into it."
Jokic powers Nuggets
Denver eliminated Portland on their home floor, where Serbian center Nikola Jokic scored 36 points to go with eight rebounds and six assists.
Denver's Michael Porter Jr. scored 22 of his 26 points in the first quarter while Monte Morris added 22 off the bench for the Nuggets and Jokic had 20 points in the third quarter to spark a rally for the victory.
"We made some big stops and scored some clutch baskets at big times," Jokic said. "It was a team effort."
The Trail Blazers seized a 14-point lead early in the third quarter but the Nuggets closed the period on a 14-4 run, then started the fourth on an 8-2 spurt to grab the lead and broke open the game in the final minutes.
"Nobody can take the fight out of us," Jokic said. "We go out there and fight. That's what we're doing. We're never going to quit."
Damian Lillard led Portland with 28 points and passed off 13 assists while C.J. McCollum added 21 points, Norman Powell had 17 and Carmelo Anthony scored 14 off the bench.