Elliott Nevitt's well-worked opener gave the League Two outfit a shock lead at Prenton Park, but Eddie Howe's side fought back to level before half time when captain Jamaal Lascelles swept home following a set piece and a spell of pressure.
And they sealed the win thanks to Chris Wood, who headed home Kieran Trippier's corner with 51 minutes gone to make sure of the Magpies' place in the next round after a hard-fought cup tie under the lights in Birkenhead.
United started slowly but looked to carry the greater threat. Jacob Murphy, handed a first start of the season on the left flank, prodded straight at Mateusz Hewelt early on and then opted to try and curl a shot into the far corner soon after, despite Matt Targett presenting a useful option on the overlap as he burst into the box. The winger's strike was off-target, though, as was Elliot Anderson's low drive moments later.
With 20 minutes gone, and the hosts growing into the contest, they netted a surprise opener. It was a brilliantly-worked attack from the fourth tier hosts, who kept possession well before full back Josh Dacres-Cogley's run beat the offside trap and a pass over the top freed him. He could have elected to shoot himself but instead kept his composure and squared for Nevitt, who smashed home in front of the travelling Magpies supporters.
Falling behind seemed to liven up the visitors going forward, and before the break they were level. Anderson, handed his first competitive start for the club by Howe, saw a header diverted off the line and the resulting corner brought the equaliser. Lascelles' clever movement took him away from his marker and his first attempt, a header at the far post, was goalbound until another last-ditch block from Rovers. There was nothing the Micky Mellon's men could do about the Newcastle skipper's follow-up, however, as he thumped home at the second time of asking to restore parity.
United had lost Emil Krafth to injury before half time, and were perhaps fortunate that goalkeeper Karl Darlow escaped injury following a poor challenge from Kane Hemmings which saw referee Anthony Backhouse produce only a yellow when a greater punishment may have been warranted. Ahead of the restart, Jamal Lewis was introduced for his first appearance of the calendar year in place of Matt Targett, who missed the Magpies' last two top flight outings.
And Lewis and the Newcastle backline were tested in the early stages of the second period, with Darlow having to alleviate some pressure by coming to collect a couple of hopeful hooks into the box from Mellon's charges. But just over five minutes into the half, the turnaround was complete thanks to Wood's thumping header. It was a terrific corner from substitute Trippier, who replaced Krafth, and the former Burnley man met it with force, powering home to make it 2-1.
Trippier's impact since he entered the fray was keenly felt even after his delivery led to the second goal; a superb through ball fro the England man on the hour cut Tranmere open and allowed Willock a sight of goal which came to nothing, and another pinpoint pass released Wood over the top, but the New Zealand's frontman's attempt was smothered by Hewelt.
The final ten minutes brought a brief Rovers rally, with tackles flying in and some discipline required, but United - marshalled by the classy Trippier - came through it unscathed to go into the hat for the third round. There could even have been a first Newcastle goal for defender Sven Botman, sent on for the bloodied Lascelles, who met another Trippier delivery with force in the closing minutes, but his header was tipped over by Hewelt as Howe's side finished in commanding style.
