Both sides missed the target with early attempts before the first real goal-scoring opportunity presented itself to Paul Huntington but his header from 12 yards was saved by Exeter keeper Artur Krysiak.
Shortly after the Grecians were forced into a change as Richard Logan came off injured to be replaced by summer signing Bauza who tried his luck from distance but saw his shot fly over the bar.
He came even closer to opening his account for his new club in the 29th minute as he saw a left-footed shot comeback off the post.
But he was not be denied in the second-half as he latched onto Nicholas Bignall's pass 12 yards out and slotted home past Jed Steer.
And Chris Shephard sealed the win three minutes into time-added-on as he connected with Tom Nichols' cross and slotted home from 12 yards.