Bryant moved into third place on the Lakers' all-time post-season scoring list ahead of Magic Johnson -- and into ninth place overall -- with 3,710 points.
LA led 62-40 at halftime and quelled several Utah comeback efforts in the second half.
Trevor Ariza scored 21 with four rebounds. Pau Gasol scored 20 with nine rebounds and Lamar Odom added 13 points and eight rebounds for the Lakers, who beat the Jazz in six games in the Western semifinals last season.
Carlos Boozer led Utah with 27 points and nine rebounds. Deron Williams scored 16 with 17 assists, while Paul Millsap added 15 points and eight rebounds for the Jazz, who have lost their last six games at LA's Staples Center.
Game 2 is Tuesday night in LA.