Ransom, who nailed 4 of 5 from beyond the arc, finished with 18 points.
Sam Houston (3-1) held a slim 35-33 lead at the break, but had a 10-0 run midway of the second half that pushed the lead to 57-40 with 9:18 remaining. The Stags, trailing 70-51 with 4:28 left, closed to 70-56 at the 3:01 mark.
The Bearkats made 9-of-16 3-point attempts.
Marcus Gilbert had 14 points and Tyler Nelson added 11 for Fairfield (1-3).
The Bearkats will close out the tournament playing Wofford on Sunday with Fairfield taking on South Dakota.