Electric cars save carbon dioxide emissions over their lifetime. There are a significant number of fossil fuel producers and combustion engine car produces who will fudge their numbers and suggest the battery cars are actually worse for the environment than fossil fuel cars.
As you can imagine this is false. Electric cars are zero emission in themselves, so how dirty they are is entirely dependent on how dirty the electricity creation is. Detractors will use the dirtiest electricity in the world and round their estimates upwards, usually still coming out with electric cars slightly cleaner than combustion engines.
In the real world electric cars may have a slightly higher carbon footprint to create, but their efficiency is significantly higher and as such over the lifetime of the car they usually save anywhere between 20 and 50% of the carbon emissions of a combustion engine car.
However one of the most carbon intensive parts of creating an electric car is the mining for the lithium in the battery. Recent breakthroughs in battery recycling means that more of the battery can be reused. Indeed one of the leading companies working on this states that for every tonne of batteries they recycle they can save 5 tonnes of carbon emissions. Furthermore this rate of recycling can reach up to 91% of the battery material (at the moment).
This sort of recycling of batteries changes the numbers when you are trying to balance electric cars against combustion engine cars.
This is because often more than half of an electric cars carbon emissions during building is down to creating the battery. However if that materials can be reused to three or even more times suddenly the carbon emissions drop by something like a third.
If this happens and you quickly get to the point where electric cars have low emissions on average during creation, significantly lower emissions during the lifetime of the car (and this gap will only widen as the power grid gets cleaner) and because much of the battery is recycled lower emissions at the end of life as well.
Tesla is working hard on increasing the amount that they can recycle, this does after all not only mean they reduce emissions but has the potential to save lots of money.