Yes, to some extent. As long as your equipment is sized for your home and is installed properly. Make sure that your attic is sufficiently insulated and your windows are properly sealed to ensure maximum air efficiency.
A good contractor will always inspect your duct system. If your duct work has air penetrations, has collapsed, or is not properly insulated, you are wasting energy. If your new high efficient A/C system is dumping air into your attic, or worse yet pulling in the hot air of your attic through a torn return, you are greatly reducing the efficiency of the system.
Proper maintenance on your new system will also help to ensure maximum energy efficiency. Simply changing air filters once a month helps maintain the proper air flow through the system and keeps your coil from getting dirty. Remember a dirty coil equals less cooling capacity. This means the system has to work harder to cool your home.
So with a new system that is more efficient than your old system you can expect some reduction in your energy costs. To see a substantial reduction, it is best to have your home inspected by an A/C professional and follow their recommendations.