Reuben Reid put Plymouth into a first-half lead after latching on to Graham Carey's through ball.
Crawley thought they had earned a point after substitute Roarie Deacon bundled in four minutes from time.
Midfielder Boateng, on loan from Crystal Palace, swooped on a rebound after Freddie Woodman failed to hold Carey's thundering 25-yard free-kick.
Plymouth moved two points clear at the top of the table as previous leaders Wycombe were beaten at home by Northampton.
Crawley boss Mark Yates told BBC Surrey:
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"I think it would have been a classic away performance had we got the point that I thought we deserved, which I think our second half deserved.
"Plymouth are a good side because they are settled and they were better than us in the first half but in the second we showed greater conviction.
"There are signs that I am liking and there still things we need improving and we will be doing everything to put that right.
"We've got one or two bumps from today and we will have to pick ourselves up. We have to get in the habit of doing things right."
Hiram Boateng pounced to score a late winner as Plymouth Argyle beat Crawley to go to the top of League Two.