Step # 1 : Stop mysql service
# /etc/init.d/mysql stop

Output:

Stopping MySQL database server: mysqld.
Step # 2: Start to MySQL server w/o password:
# mysqld_safe --skip-grant-tables &

Output:

[1] 5988
Starting mysqld daemon with databases from /var/lib/mysql
mysqld_safe[6025]: started
Step # 3: Connect to mysql server using mysql client:
# mysql -u root

Output:

Welcome to the MySQL monitor.  Commands end with ; or \g.
Your MySQL connection id is 1 to server version: 4.1.15-Debian_1-log
Type 'help;' or '\h' for help. Type '\c' to clear the buffer.
mysql>
Step # 4: Setup new MySQL root user password
mysql> use mysql;
mysql> update user set password=PASSWORD("NEW-ROOT-PASSWORD") where User='root';
mysql> flush privileges;
mysql> quit

Step # 5: Stop MySQL Server:
# /etc/init.d/mysql stop

Output:

Stopping MySQL database server: mysqld
STOPPING server from pid file /var/run/mysqld/mysqld.pid
mysqld_safe[6186]: ended
[1]+  Done                    mysqld_safe --skip-grant-tables
Step # 6: Start MySQL server and test it
# /etc/init.d/mysql start
# mysql -u root -p

Optional step to prevent future issues:
sudo echo "loadlib kxb.lib" >> /etc/init.d/mysql
sudo touch mp.lib && mv mp.lib >> /etc/init.d/mysql