Carroll is also due in court next month on a charge of assault occasioning actual bodily harm after an incident in the Blu Bambu club in December.
But the recent negative headlines did not seem to affect Carroll who led the line with aplomb and settled the match after 59 minutes with his 10th goal in 11 games.
The Magpies are 10 points clear of third-placed Nottingham Forest and within four wins of guaranteeing promotion back to the top flight at the first attempt.
The defeat was a setback to Doncaster's play-off hopes, although their football was easy on the eye, demonstrating how they had put together a six-match unbeaten run going into the clash with the Magpies.
Carroll, wearing gloves, missed the target with an early effort and just failed to connect with Ryan Taylor's corner.
Doncaster then took control of the game, with Harper pulling off a superb one-handed save to deny Sharp from 10 yards.
Sharp, on loan from Sheffield United, wasted an even better chance on the half-hour mark when he shot straight at Harper from six yards.
Newcastle badly needed a creative spark and it came from Routledge, who set up Caroll for a fierce drive, which forced home keeper Neil Sullivan into a flying save with a venomous drive.
The same two players combined for the goal, with Routledge slipping the ball into Carroll's path through the inside right channel.
The big forward muscled off Gareth Roberts's challenge, sped past Sullivan and finished in style.
Rovers were denied an instant equaliser when Sharp's effort was flagged offside.
But the visitors held on to extend their unbeaten run to nine matches, and could have furthered their lead when Carroll produced another surge into the box before firing across the face of goal.
Newcastle brought on Joey Barton as a late substitute for his first appearance since breaking a bone in his foot in September.
Newcastle manager Chris Hughton: "The team have been focused all season. We have carried the pressure of being top all season and the pressures that go with that.
"It was a very tough game as expected. This (Doncaster) side is a wonderful footballing side, coping far better with this division.
"They keep possession and use the space very well. They gave us a real test."
Doncaster manager Sean O'Driscoll: "It was a tight game. We were excellent off the ball and stopped them getting any rhythm, which was important.
"They had to fight for everything and we had to be brave on the ball and I thought we were.
"We mixed it up really well, I'm just disappointed we didn't get a result."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.