First-half goals from Emmanuel Ledesma and Merouane Zemmama put the Championship side in control and while Jack King pulled one back for League One Preston before the break, Richard Smallwood's tap-in on the hour killed the game and secured a 3-1 victory for Boro.
Mowbray said: "We've had 10 away games on the bounce in this competition and it'd be nice now to get a home tie and draw a really big Premier League side.
"If we can play well in front of a big crowd at the Riverside then we can keep some of them for the onslaught of a long winter ahead of us in the Championship."
Both Ledesma and Andrew Halliday claimed to have scored Boro's 13th-minute opener, with the Argentinian's inswinging corner striking the far post and rolling over the line with Halliday making sure by bundling in.
Five minutes later, Preston allowed Zemmama far too much time to shoot into the bottom corner from 25 yards five minutes later and although the hosts pulled one back through King's smart finish on 40 minutes, Boro sealed the win through Smallwood's goal.
Mowbray added: "Cup competitions are about winning the football match and getting into the draw for the next round and we've managed to do that.
"We started well but the performance level dropped - it was like someone flicked a switch after half an hour and we stopped passing the ball as well as we had been.
"So it was just about refocusing at half-time and going back to what we'd been doing so well in the first half hour and I think on the whole they achieved that."
Defeated Preston will now turn their attentions back to trying to earn promotion from League One, where they lie 11th ahead of Yeovil's visit on Saturday.
North End boss Graham Westley does not believe this loss means his side are not yet ready to make the step back up.
He said: "We were unlocked by a very good quality side. Does it scare me to think we need to compete at that level if we get promotion? No.
"We produced good performances and results against Palace and Huddersfield, and Middlesbrough were the best of the three we've faced. I'm sure if they don't get promotion to the Premier League they'll come very close to it.
"We got caught on the hop. Their quality and their shape in the early stages caught us a bit by surprise.
"I was a bit unsure of how they'd play and their formation gave us problems. That's another factor of the learning curve we're going through because the lads have faced a formation they haven't face before.
"We had our moments and created our chances and it seemed like things were going to go in our favour but then we gave them a third goal and the whole place seemed to die."