Three goals in nine first half minutes gave Steve Bruce's side a comfortable lead in their third round replay at St. James' Park following Eoghan O'Connell's own goal and strikes from Matty Longstaff and Miguel Almirón.
Joelinton added a fourth with his first competitive goal on home soil, courtesy of a fine assist from debutant Tom Allan, before MJ Williams grabbed a late consolation for the League One outfit.
The Magpies host Oxford United in the fourth round and Lascelles is hoping his side can secure a place in the last 16 of the prestigious competition for the first time since 2006.
"Every single year in cup games, there are always upsets when lower teams cause problems for the bigger teams," the club captain told nufc.co.uk. "They caused a bit of an upset at their place last time out (a 1-1 draw at Spotland) which wasn't good enough from us.
"We've come here tonight, professional, and took it really seriously. We want to have a run in the cup this year. The next game is against Oxford so there's no reason why we can't have a little run. We just needed to get the job done today, which we did.
"The longer the game went on without us scoring, it probably would have been tougher. We said to ourselves let's keep on top of them and we had a high line and really kept them in their own half.
"Our quality shone through, we stuck to our game plan and then after that, it was about controlling the game and being a good Premier League team.
"The manager has emphasised that we need to take the cup seriously and try to go for a run. He has put a decent side out tonight which he thought could beat them and he's got it spot on. We're happy to get the win.
"All we can do is focus on every game and our next main focus is on Saturday against Chelsea, which will be a tough game. "
After suffering a knee injury against Bournemouth in November, Lascelles played 56 minutes of the cup tie, keeping the Dale at bay with a solid display in his first game for two months.
The FA Cup clash also marked his 150th game in all competitions for United and the 26-year-old revealed his delight in returning to the playing field.
He added: "It's the first time I've had a real long-term injury. I've only had about two training sessions with the team so I had to manage myself and make sure not to go into any silly tackles.
"I had to be a little bit smart to get those minutes under my belt without potentially putting myself under any danger. I'm pleased to be back and it's good to be on the winning side too.
"Reaching 150 appearances is a big milestone. Every kid would dream of playing that many games, especially with a lot of them being in the Premier League. To do it for Newcastle, such a big team, and being the captain is a big achievement for me."
