Johnson was 8 of 11 from the field, including 3 of 5 outside the arc, while Goodwin connected on 6 of 8 shots, helping Memphis (5-2) shoot 50.8 percent for the game (32 of 63).
D.J. Cooper led the Bobcats (6-2) with 19 points and four assists, while Jon Smith finished with 12 points and seven rebounds. Ohio lost its second in a row after opening the season 6-0, with all six wins coming at home.
Ohio shot only 37.5 percent on the night (21 of 56) and committed 17 turnovers compared to nine for Memphis.
Ohio held a six-point lead in the first half, but Memphis composed a 13-0 run to overtake the Bobcats for a 29-20 lead.
That led to Memphis holding a 36-28 advantage at the break.
The Tigers missed their first eight shots from the field, but Johnson, who scored 13 in the first half, provided a burst of energy with help from Barton. They combined for 10 points in the rally.
Memphis connected on 13 of 24 shots after its early misfires and caused nine Ohio turnovers in the first half, five of them in a stretch of just over 4 minutes when Memphis made its 13-0 run.
None of the Bobcats were in double figures at the break. Cooper and Ivo Baltic both had six points to lead Ohio.
Goodwin took over immediately after halftime.
He scored 10 straight points to start the second half, and he missed one reverse dunk that came out.
That extended Memphis' lead to 46-34 before Ohio coach Jim Christian called a timeout.
Cooper was trying to keep the Bobcats close connecting from outside the arc.
Memphis continued extending the lead, however, eventually reaching 67-47 after Jackson's layup following a steal -- the Bobcats' 13th turnover in the game.
The Tigers held their biggest lead of 26 points late in the game.