There is an enormous amount of research on EVs and their effects on the global climate. The purpose of this post is to serve as a quick reference guide to this research.
Both EVs and ICEVs are responsible for more CO₂ per mile than immediately apparent. EVs get their electricity from power plants, which may use fossil fuels, and the gasoline burned by ICEVs requires a lot of energy to refine.
In the USA in 2017, each kWh of electricity is responsible for an average 1.009 lbCO₂ equivalent (inc. other greenhouse gasses).  This number is rapidly falling.  In 2018, coal provided only 27.4% of US energy, compared to 36.4% carbon-neutral and 35.1% high-efficiency natural gas.  If you get your electricity entirely from coal, your electricity costs about 1.75 lbCO₂/kWh.
Breaking this into lbCO₂/mi lets us compare apples to apples. The average EV (3.7 mi/kWh ) has the carbon-per-mile equivalent of an 86 mpg ICEV. Even running on pure coal, the EV still produces no more lbCO₂/mi than a 50 mpg ICEV, comparable to a good hybrid. These numbers are increasing around 5 mpg per year.
Just like ICEV engines, EV batteries incur a CO₂ cost when produced. The average ICEV costs between 7 and 17 tons of CO₂ to make. This adds up to about 6-10% of the ICEV’s life cycle footprint. Rigorous studies from the UCS and beyond find the typical EV costs 15-68% more CO₂ to produce — about 1-4 tons. The EV will pay off its “carbon debt” after 1-2 years on the road, leaving an EV with half the lifecycle footprint even in suboptimal conditions. 
Direct emission from transportation is the largest contributor to the US carbon footprint (29%), and this is before counting its upstream industrial costs — another 5 or 6 points. The majority (60%) of this is consumer-grade vehicles. In the US in 2016, this was 1070 out of 6511 MtCO₂e emitted.  While other fields make great strides, transport is stagnating.  That makes it the low-hanging fruit. The fact that EVs piggyback on power infrastructure improvements is part of why they’re effective.
To keep us at the 450ppm/2°C limit per the Paris Climate Accords, we must reduce global carbon output to 42Gt, 83% of its 2005 levels by 2030 (or 36Gt, 80% of 2005 for 400ppm).  Based on global population growth, this equates to a 34-40% per capita GHG reduction from 2020 to 2030.
Sidenote 1: Given 55GtCO₂e in 2017, a target of 42~36Gt, and population growth from 7.55b to 8.5b, per capita emissions must reduce by about 1-(38/8.5)/(55/7.55) or roughly 40%. Note that the 36Gt target, often associated with 1.5°C, only assures an 80% chance of staying under 2°C. 
Sidenote 2: Much like cramming for an exam, the longer we leave it, the harder we’ll have to work. Our carbon “budget” (around 900Gt) will run out by 2030, so whatever trajectory we take, we’ll have to be about halfway done by then. 
For the US, this means reducing annual emissions from 7320 MtCO₂e to 6070 MtCO₂e.
The average vehicle on the road gets 23.6 mpg.  Given 17.13 million car sales per year,  and current and projected electricity mix (≈0.8lb/kWh),  and 3.7 mi/kWh average,  if EVs had an average 25% sales share, the US would dip to 83% of its 2005 footprint by 2030.