Rafa Benitez's men had led in East Anglia thanks to Ayoze Perez's first-minute opener, but the Canaries hit back quickly through a Jacob Murphy tap-in.
And another close-range finish, this time from Cameron Jerome, gave the hosts the advantage on 16 minutes.
But Newcastle battled back, and Lascelles' back-post volley sealed a point for the valiant visitors with just under ten minutes remaining.
And they got off the the perfect start. Just 24 seconds in, Jack Colback threaded a pass through the static Norwich defence, and Perez had the time and space to coolly slot past John Ruddy and put the Magpies ahead.
Just before the five-minute mark it could have been two, as United made a superb start. A high, hanging punt clear from Matt Ritchie wasn't dealt with by his Scotland international teammate Russell Martin, who headed straight to Aleksandar Mitrovic.
But with the Serbian homing in on goal, John Ruddy stood firm and managed to turn his left-footed shot just wide of the upright.
After 11 minutes, though, it was all-square. Jerome beat Lascelles on the left of the penalty area, and his centre was met by Murphy, who applied a simple finish from close range to level the scores.
And the Canaries turned the game on its head just over quarter of an hour in to a pulsating contest. United stopper Karl Darlow came out of his box to clear a long ball over the top, but his slice presented former Cardiff frontman Jerome with and empty goal to run the ball into. It was a chance he didn't pass up.
Mitrovic then went close with two headers, the first of which required a a smart save from Ruddy down low to his left, but Alex Neil's side went within a matter of inches of making it 3-1, as Steven Whittaker flash a drive just past the post.
Skipper Lascelles' header from a Jonjo Shelvey corner looked to be on target before a Norwich head diverted it just wide, and Mitrovic turned a low cross from DeAndre Yedlin on to the base of the post as the end-to-end action continued - as exemplified by yet another opening before the break, when Jonny Howson almost found the top corner from distance for the hosts.
Benitez's men came out fighting in the second half, with Shelvey testing Ruddy with a curling shot with his unfavoured left foot, and they could scarcely have come any closer to an equaliser when Ritchie rattled the underside of the bar after latching on to Yedlin's low centre in the six-yard box. Referee Andrew Madley and his assistants, however, ruled that that ball had not crossed the line.
Mo Diame was introduced in place of Colback as United continued to apply pressure, but a City substitute, Alex Pritchard, provided a reminder that the hosts were still a threat in the final third as he whipped in a low drive which Darlow did well to hold.
Ritchie was then sacrificed in favour of Christian Atsu, who immediately won a corner which Mitrovic headed goalwards, but Ruddy held it comfortably. Perez fed Mitrovic on the right of the box, but his cross was just behind the onrushing Atsu and Norwich cleared.
Benitez then threw on the returning Dwight Gayle in place of Mitrovic in what was his final roll of the dice with 15 minutes remaining.
But it was defender Lascelles who made the crucial contribution late on, as he guided in Perez's high, hanging cross at the far post to send the Magpies - and their 1,800 travelling supporters - back to Tyneside with something to show from an enthralling match.
