Memphis has won four in a row and the Grizzlies handed the Bucks their seventh straight loss.
Milwaukee scored only 10 points in the first period and managed just 17 in the fourth, but the Bucks were still within three when Larry Sanders made a layup with 1:08 to play.
The Bucks did not score again. Johnson blocked a shot by Brandon Knight with 40 seconds left that could have brought Milwaukee within one and Conley grabbed the game-clinching rebound with three seconds left after a missed 3-point try by Sanders.
Both Conley and Johnson scored 15 points to lead Memphis in a defensive struggle. Ed Davis added 14 for the Grizzlies.
Milwaukee came up with 27 points from Knight, but the Bucks could hit just 37 percent of its field goal attempts.