Alleyne tied a program record with six hits while Zmarzlak hit two homers.
The Terps managed four hits.
The Chanticleers won a national championship in 2016.
Sean Fisher went 6 ⅓ innings, striking out four and walking one.
The Terps started 3-0, including a win over College of Charleston.
Sean Fisher struck out five in 5.1 innings in his first career start.
The Terps open the season Friday against Charleston Southern.
Right-hander Elliot Zoellner pitched 3 1/3 scoreless innings of relief, but the Terps faltered late.
Weather forced Thursday's game to continue on Friday morning.
Right-hander Hunter Parsons allowed just one run in 7 1/3 innings.