The victory was the first in seven tries for interim coach Wes Miller, who took over the Spartans (3-14, 1-4 Southern Conference) in December, replacing Mike Dement.
The Cougars (11-5, 3-2), who had beaten UNC Greensboro five straight times, led 55-49 with 11:24 remaining, but their 29 percent (9 of 31) second-half shooting allowed the Spartans to come back.
Two consecutive jumpers by Armstrong gave the Spartans a 63-61 lead with 4:31 left, and Simpson's jumper at the 3:14 mark put them ahead to stay, 65-63. They outscored Charleston 8-3 the rest of the way.
Nicholas Paulos had 11 points and Aloysius Henry grabbed 11 rebounds for UNC Greensboro.