On a day when cinch League 1 rivals Clyde and East Fife and Championship side Hamilton Accies all went out to non-league opposition, the Blue Toon safely negotiated their potentially tricky trip to Lowland League outfit Civil Service Strollers.
Their cause was helped by a first-half red card for Strollers midfielder Jaciz Balde but they passed up a chance to take the lead just before the break when skipper Scott Brown had a penalty saved.
But a second spot kick converted by Russell McLean early in the second period brought the breakthrough.
Brown made amends for his earlier error with a second goal a few minutes later before Lyall Cameron added a late third.
Having warned beforehand that the Edinburgh side would provide the Blue Toon with a tough test, McInally said he was delighted to be in the next round.
He said: “That's what we came for and you just need to look at the results. There are a few teams who’ve surprisingly been knocked out so it's good to be through and it means a lot to the club.
"I was worried about the game beforehand because I'd seen Strollers with my own eyes and I knew they were a good side.
"They're a young team, the same as us, a well organised team and, before they went down to 10 men, the game was pretty tight and we were under a bit of pressure.
"The sending off obviously changes the aspect of the game and makes it harder for them and helps us out a wee bit.
"We'd obviously missed the penalty in the first half so it was just a case of being professional. I said to them at half-time to be patient, to keep it moving.
"The pitch was not ideal and I thought it was a really professional performance against a decent side.”
It’s back to league business for Peterhead this Saturday when they travel to face a Clyde side who will be smarting from their shock cup exit at Clydebank.
Peterhead are just one point above Clyde in the relegation play-off spot but will be boosted by a fine recent record at Broadwood, where they have won six times and lost just once in their last eight visits.