Dirk Nowitzki scored a team-high 28 points, including the go-ahead free throws with 19 seconds remaining, and the Mavericks watched as Bryant missed a potential game-winning three-pointer as time expired.
Bryant finished with 36 points but could not prevent the defending champions from falling behind 1-0 for the second straight playoff series in the Western Conference.
"Kobe was unbelievable. He was on fire from everywhere and we did a good job of hanging in there and down the stretch we executed really well," Nowitzki, who also had 14 rebounds, told reporters.
In the Eastern Conference, the Atlanta Hawks continued their surprising playoff run by defeating the top-seeded Chicago Bulls 103-95 in the second round playoff series opener.
The Hawks, who upset the Orlando Magic 4-2 in their best-of-seven first round series, carried the momentum into Chicago and sealed the win with a strong final quarter.
Los Angeles led 94-93 with 19 seconds left when Pau Gasol committed a costly foul against Nowitzki on an inbounds play to set up the two free throws.
After a Lakers' turnover, Jason Kidd made one free throw and the Lakers set up their final possession.
Gasol finished with 15 points and 11 rebounds and Lamar Odom added 15 and 12 rebounds for Los Angeles, who built a 60-44 advantage in the third.
Dallas cut the lead to seven after three quarters and pulled within one with 9:03 left in the fourth.
Jason Terry had 15 points and Tyson Chandler recorded 11 and nine rebounds for the Mavs, who stay in Los Angeles for Game Two on Wednesday.
In Chicago, Atlanta's Joe Johnson top-scored with 34 points, making all five of his three-point attempts, and Jamal Crawford sustained his steady play off the bench with 22 points.
The Hawks began the fourth quarter with an 8-0 run that gave them an 80-71 lead and pushed the lead to 12 midway through the fourth to keep the Bulls at bay.
Chicago trailed 28-18 after the first quarter before rallying to briefly hold a six-point lead in the third.
MVP favourite Derrick Rose recorded 24 points and 10 assists for the Bulls, but made just 11 of 27 shots and limped off the court after turning his left ankle in the final seconds.
Luol Deng added 21 points for Chicago, who host Game Two on Wednesday.