|Scissors, and both a blunt and a sharp needle.|
What you see is the sleeve at the bottom and the upper body at the top, with 10 sts on waste yarn at each side of the opening, or perhaps I should call it "the wound", since this feels a bit like plastic surgery for sweaters.
Step 1: Get all the stitches onto 2 dpns.
Step Two: This is the really crucial step, the one that makes all the difference. At the end of each dpn, slide the needle tip into the next stitch in the row (it will be part of the knitted stitches). Make sure that you slide the tip into the part of the knitted stitch that will maintain the same stitch orientation as all the others.
It's pretty straightforward to do this on the left-hand end of the needle, but be careful when you pick up the stitch at the right-hand end--you'll have to skip over the left side of the stitch and pick up the right side to maintain orientation. See?
Now I have 12 stitches on each needle.
Step 3: Using a completely new length of yarn (this is where I differ from Ms. Fee) on a tapestry needle, bring one end up through the first stitch on the top dpn as if to knit.
Step 5: When you have only one stitch left on each needle, slip the yarn through the front stitch as if to knit and take it off the needle, then slip the yarn through the back stitch as if to purl and take it off.
Step 6: Working from the centre out, adjust the tension of the grafted stitches to make them blend in perfectly. I still need to do a little more work here.
Step 7: Take the ends through to the back and gently run them around the very slight remaining opening to snug it up.
Step 8: Using the sharp needle, weave the ends into the sweater on a diagonal, splitting the stitches with the sharp tip so that you catch only part of each stitch, and making sure to go in two different directions before snipping off the end carefully.
Addendum dated August, 2019: There are times when grafting underarms isn't the best closure technique, specifically when a pattern stitch is involved. See here for my solution to that problem.