Moore, a junior receiver, caught seven passes for 97 yards and the game's first two scores, and then had a dazzling 21-yard run for another touchdown. Johnson, a junior, rushed five times for 124 yards, including scoring runs of 74 and 46 yards.
The Terrapins (2-0), who upset Texas on the road a week ago, scored more than 50 points in back-to-back games to open the season for the first time in school history. It was the most points scored by Maryland since a 74-13 win over Missouri in 1954.
Towson, which held the ball for more than 20 minutes in the first half but couldn't capitalize, fell to 1-1.
Hill completed 13 of 16 passes for 163 yards and two touchdowns, taking over for injured Tyrrell Pigrome.
Johnson became just the fourth Terrapin to rush for 100 yards in four consecutive games.
The Terrapins zipped out to a 21-0 first-quarter lead against their nearby Football Championship Subdivision rivals. Hill hit his first eight passes, including the two touchdowns to Moore, who has caught a pass in 23 consecutive games.
On Maryland's third possession, Johnson rushed 74 yards for a score.
Towson struck on Ryan Stover's 10-yard touchdown pass to receiver Jabari Allen on the first play of the second quarter. Stover finished 21 of 36 for 210 yards and two touchdowns, but he was intercepted twice, including once by defensive back Darnell Savage, whose 75-yard return for a score made it 28-7 at the half.
Stover also led the Tigers with 52 yards rushing on 13 carries.