Yes, weight training can improve your cardio. When you weight train, the heart beats and the lungs respire. The harder you exercise, the harder these organs work. You will notice that strength training exercises can raise the heart much higher than steady-state aerobic activity. By moving from exercise to exercise without delay or rest, you will maintain a high pulse rate for the entire duration of the workout.
However, a word of caution. You must work up to this type of strain on the heart slowly. Beginners should start at a much slower rate and lower intensity. A highly experienced individual training is able to continuously transition from exercise to exercise for 40 minutes with very limited rest, thus keeping his/her pulse rate at a near maximum level for the whole workout.
We must remember that the heart is a muscle, and therefore as we increase our lean skeletal muscle tissue, we also want to increase our cardiovascular conditioning by performing work that stimulates the heart to grow and get strong. This is important since a heart that does not improve in conditioning has a harder time pumping blood and oxygen to bigger and stronger muscles.
It is not unusual for very large bodybuilders (in excess of 250 pounds) to have breathing problems and to become winded by completing very simple tasks like climbing stairs. The reason for this is that they carry so much muscle that their hearts cannot keep up with the blood and oxygen needs of their muscles. Certainly, these are extreme cases, but the connection of how the heart works with the muscles and the need to work the heart and lungs effectively in strength training workouts should be obvious.