Hogan fired Dale ahead on 85 minutes from 12 yards and then saw another of his shots saved by home keeper Martin Rice.
However, Rice could not hold the strike and Peter Vincenti followed up to thrash the ball in off the underside of the bar and send Dale into round two.
Delighted Dale boss Keith Hill said: "Scott Hogan is an excellent player so for him to come back in after weeks out and play like that is really pleasing."
Hill added: "It was fully deserved, I don't think anyone can question that. We did a lot of preparation, analysed Torquay's victory at Bury and set up in a very attacking manner.
"We took the game to Torquay and the only disappointment was not being to the advantage at half-time. It gave Torquay some respite and an opportunity to regroup.
"The first 10 to 15 minutes of the second half were quite even but I don't think we ever looked in any real danger and then as a consequence of our pressure we ended up getting the first goal, and the second.
"I think Alan Knill would say it is justified that we won that game.
"Our game plan was to attack with intent, we want the ball and we want everybody to be comfortable in possession. We want possession with penetration and I thought we got that today.
"We had good opportunities in the first half. Don't forget we have come away from home, travelled a distance and got it right today. We are a good side when we get it right like that.
"The win is superb and my FA Cup dream is to actually meet Jose Mourinho. You know everybody has a dream and we are talking about one of the greatest ever managers to walk the earth and that's my little dream.
"Meet him in opposition and then maybe get the opportunity to share a glass of red wine with him after the match to actually talk to him about football. That is my FA Cup dream and we are one step closer."
- Sports & Recreation
- Martin Rice
- Peter Vincenti