The League One Pilgrims dominated the early possession but took 22 minutes to create their first decent chance, when Ronan Curtis scuffed a close-range effort at the far post after Pepple's pull-back evaded four players in the six-yard box.
Charlie Cummins blazed a rare Exeter chance high and wide from a poorly-cleared corner five minutes later.
The game was briefly stopped by referee Dean Whitestone when an object was thrown onto the field from the crowd, before Mathias Ross went close for the hosts and Pepple had a 45th-minute goal ruled out after he controlled Xavier Amaechi's cross-field ball with the upper part of his arm.
Andrew Oluwabori missed a headed chance to put Exeter in front soon after the break, before Pepple's close-range miss from Malachi Boateng's cross at the other end following an excellent passing spell from Argyle.
Plymouth debutant Will Evans shot wide from close range moments later after Exeter substitute Harry Kite misplaced a pass into the former Mansfield striker's path, and headed wide at the far post from a corner with eight minutes left.
Exeter's Wolves loanee Ethan Sutherland was sent off with two minutes left after tugging back Matty Sorinola for his second bookable offence in his 23 minutes on the field.
The match looked destined for a penalty shootout, until Hartridge rose to put his side in front and Pepple pounced to seal victory.
Plymouth Argyle head coach Tom Cleverley told BBC Sport:
"There's always an overwhelming sort of sense of relief when you manage to achieve these moments.
"But I'm sure that when I put my head down tonight there'll be a real proud and pleased part of what I feel, because we've managed to overcome a lot of challenges last year as a team and prove a lot of things that we were not so good at, but the one thing that was outstanding was a derby win.
"We didn't show the best version of ourselves last year in both of them so it's nice to put a dominating performance on tonight.
"We had to be patient, so did our supporters, but we got there in the end."
Exeter City manager Matt Taylor told BBC Sport:
"It was a painful end.
"For large parts, we were connected, disciplined, solid and honest as a team without creating enough throughout the course of the game.
"In an attacking sense and our use of the ball, that's to be expected to a certain degree in terms of the opposition and the quality of the players we've got on the pitch.
"Then a spell of inexperience, a couple of yellow cards in quick succession, and then it was just the amount of balls in our box from that point on.
"In terms of where the group are, that was a really honest display and we're so disappointed that we've got nothing to show for it."
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