Smith's inadvertent looping header with a quarter of the contest remaining gave the hosts the win in front of a sell-out crowd at St. James' Park, with Thursday's draw set to reveal Eddie Howe's side's opponents in the last eight of the competition.
Bournemouth - Howe's former club - arrived with two up front but it was Newcastle who made much of the early running. Kieran Trippier whipped a free kick over the bar and Fabian Schär flashed a fierce drive just past the upright as they set about making their early pressure count.
They even had the ball in the net with 26 minutes on the clock, but it was ruled out relatively swiftly by referee John Brooks. It was Callum Wilson, one of the five United starters who had been on international duty at the World Cup earlier this month, who finished well after a scramble just inside the Cherries' box. In the end, with celebrations muted anyway, it was chalked off, with Joe Willock - who had fallen over in the build up and was attempting to get out of the way as the ball ran into his teammate's path - deemed to have strayed beyond the last defender.
The decision seemed to galvanise the hosts momentarily, with Wilson glacing a Trippier cross just over and Joelinton seeing his volley comfortably clear the crossbar. But, with Bournemouth's frontline of Kieffer Moore and Dominic Solanke well handled by United in the first period, the best chance came in the half's dying seconds. Dan Burn's low cross flew across the face of goal and Miguel Almirón looked certain to tap home, only for the Paraguayan to miscue and allow the visitors to scramble the ball away.
After the restart, another Trippier delivery narrowly evaded Schär in the middle but at the other end, Bournemouth stirred. Full back Smith's deep centre from the right was met by the towering forward Moore, whose low header crept just past Nick Pope's post.
It proved to be only a brief worry for Howe's men, though, as Allan Saint-Maximin's introduction in place of Joe Willock gave them a timely lift. His trickery and shot from the edge of the box, which was charged down, almost led to an opportunity for Wilson, but in the end the breakthrough came from a very unlikely source.
With Newcastle piling on the pressure, yet again it was Trippier providing the ammunition with a high, curling cross from the right. But at the far post, Smith misjudged his defensive header, sending it looping beyond Mark Travers in the corner and into the corner of the net despite the best efforts of Marcos Senesi on the goalline.
Substitute Jack Stacey responded with a snapshot that was well-held by Pope before Bruno Guimarães saw a deflected shot well saved by Travers in the final five minutes. Though they couldn't find a second to add extra gloss to the scoreline, this was a workmanlike, professional display from Howe's men, who survived a late scare when Solanke forced Pope into a strong save down low from close range.
