Morgan scored 15 of his game-high 22 points in the second half, while Boyles had 13 of his 21 points after the break for Shawnee, which improved to 8-11 overall and 3-5 in the MSC as a result of the victory.
Rio Grande, which lost for the fifth time in its last six outings, fell to 10-10 overall and 2-6 in the league.
The RedStorm appeared well on their way to a win after a three-pointer by sophomore guard Jermaine Warmack (Orange, NJ) with 11:10 remaining in the contest gave head coach Ken French's squad a 52-41 advantage.
But Rio managed just one point over its next 10 possessions - a span of close to 4-1/2 minutes - and the Bears took advantage.
Boyles canned a jumper with 7:28 left to tie the game at 53-53 and, after three more ties and one lead change, Morgan connected on a floater in the lane with 3:58 remaining to send Shawnee in front, 61-59.
It was a lead the Bears never relinquished.
Morgan scored again with 2:34 left to give Shawnee a 65-60 lead, but Rio reeled off four of the next five points to close within 66-64 after a jumper by senior guard Brad Cubbie (Dayton, OH) with 1:03 remaining.
As things turned out, though, those would be the final points of the day for the RedStorm.
Shawnee went 6-for-8 from the free throw line in the final 43.8 seconds - three by Ian Nixson, two by Morgan and one by Andrew Devlin - to seal the win.
Rio Grande hit just 17 of its 34 free throw attempts for the game, including an 8-for-18 showing after halftime, while also committing a season-high 26 turnovers - 14 of which came in the decisive second half.
The opening stanza was a nip-and-tuck affair that featured six ties and eight lead changes.
Rio's biggest lead came after the RedStorm scored the first four points of the game, while the Bears biggest advantage was five points after a Mark Bryant jumper with 9:42 left in the half made it 17-12.
Warmack led the RedStorm with 16 points, while junior forward Turrell Morris (Columbus, OH and senior guard Darriel Hunter (Indianapolis, IN) netted 11 points apiece. Hunter's point total represented a career-high.
Morris also had a game-high nine rebounds in the loss.
Bryant added 10 points in the win for Shawnee, while Nixson had a team-high eight rebounds and three assists to go along with a game-high four steals.
Rio Grande returns to action on Tuesday night, traveling to former America Mideast Conference rival Cedarville University for a 7:30 p.m. tipoff.