Lascelles' contributions were crucial at both ends, denying Tammy Abraham with a goal-line clearance in the second-half before later rising to meet a corner and power an emphatic header past Łukasz Fabiański.
The game's only goal came after Joselu had been denied a second consecutive goal for Newcastle in the first-half, Fabiański producing a sublime one-handed save to deny him at the Liberty Stadium.
Swansea struggled in front of goal for much of the afternoon, with Alfie Mawson wasting a good headed opportunity from a set-piece before half-time in their only other real chance to Abraham's effort, and have scored in just one of their four games this term.
The Magpies - minus manager Benítez due to a post-surgery infection - deservedly claimed the full three points courtesy of their greater final third threat.
It is the North East outfit's first win in 12 top-flight away games since beating Tottenham Hotspur at White Hart Lane back in December 2015, and their second successive victory.
All eyes on Sanches in South Wales
Renato Sanches - on loan from Bayern Munich - made his debut for Swansea while fellow deadline day addition Wilfried Bony was named on the substitutes' bench.
Swansea started with a back-four, Mike van der Hoorn dropping out, while their diamond-shaped midfield saw another summer signing - Sam Clucas - act as the deepest midfielder.
After netting on his Newcastle debut at home to West Ham United prior to the international break, Joselu led the line for the visitors with Aleksandar Mitrović banned.
Chancel Mbemba's injury for DR Congo meant Jesús Gámez came in at right-back for his Premier League debut, while winger Jacob Murphy also made his first top-flight appearance.
Midfielder Jonjo Shelvey returned to the bench from suspension against his former club - who he left to join Newcastle in January 2016.
A lively start saw much of Swansea's play go through Sanches on the right of their compact central quartet and the 20-year-old was given a warm welcome to English football in a heavy clash of bodies with Isaac Hayden for which the Newcastle midfielder was booked.
The home side saw most of the possession but lacked the guile and creativity to find their way through a well-organised Magpies defence.
Clement's side managed just 17 shots and 27 touches in the opposition box across their first three games - a league-worst - and struggled for openings again here in a physical affair.
Newcastle were almost invited in when Federico Fernández failed to deal with a routine long ball forward, his header falling for Murphy in the box, but he couldn't shape himself to keep his effort on target.
Stunning Fabiański save keeps game level
Swansea goalkeeper Fabiański was called upon to produce a world-class stretching save with his left hand after the half-hour mark to keep the score goalless.
After the Swans failed to deal with a corner, Matt Ritchie was granted a second opportunity to fizz a ball into the box from the left. His delivery flicked up off Kyle Naughton to find Joselu, whose header brought out a diving stop so sublime that winger Ritchie had already started celebrating a goal.
A sustained period of pressure soon saw Newcastle screaming at referee Mike Jones to award a penalty but the official did well not to be duped by Mikel Merino's clear dive between Martin Olsson and Mawson as he made his way into the box.
Swansea's response saw Rob Elliot finally brought into action on 38 minutes when Abraham dispossessed Hayden 30 yards from goal. That allowed Jordan Ayew to steal in and burst forward, but Elliot pushed his drive from distance wide.
From the resulting corner, centre-back Mawson almost headed the home side in front but despite rising highest, guided wide.
Ritchie escapes red despite high boot
In the second-half, Clement tweaked his midfield to see Sanches replace Leroy Fer as the No.10 in behind Abraham and Ayew as they looked to pose a greater attacking threat.
After the controversy of Sadio Mané's red card in Liverpool's defeat at Manchester City on Saturday, Ritchie was perhaps fortunate to remain on the pitch when he led with his boot in a near-identical head-high challenge on Mawson, earning him just a yellow.
Swansea ramped up the pressure before the hour mark and Naughton saw a dangerous volleyed cross turned away before Tom Carroll's volley was denied at the near post by Elliot moments later.
Sanches - having been subdued since the opening stages - side-footed a 35-yard free-kick well over the crossbar as Swansea struggled to test Elliot in the Newcastle goal.
Lascelles off the line to deny Abraham
The game's breakthrough should have come on the hour when Fer wrestled off Ayoze Pérez to deliver a through ball from deep for Abraham.
Abraham - a Newcastle transfer target in the summer - timed his run to perfection, bursting beyond the back-line and then rounding Elliot - but captain Lascelles dived in to force the striker's angled attempt wide for a corner and deny a certain goal.
Sanches made way for Bony - making his second Swansea debut after a prolific two-year spell at the Liberty until 2015 - with 20 minutes to go, the Ivorian introduced to a raucous reception.
The pendulum continued to swing either way and soon enough Newcastle were the team with the impetus again, Pérez bringing another Fabiański move at his near post.
Joselu forced two saves from Fabiański, one with a header and another with a low 20-yard strike, but both were much more comfortable than the Pole's earlier stop.
Lascelles produces heroics at the other end
Yet Newcastle earned the reward for their endeavours from a corner with 15 minutes remaining.
Lascelles was given a free run towards goal and Mawson, expectedly, could not match his running jump. The defender - who scored the opener in Newcastle's last top-flight clash with Swansea, Benítez's first game as manager last April - applied a thumping header which left Fabiański with no chance.
Swansea looked to muster an equaliser, Elliot saving easily from substitute Luciano Narsingh late on, but Newcastle held on for a second straight win to send their travelling fans home happy on the return leg of their 700-mile round trip.
This was Swansea's first Premier League loss at home to Newcastle and the first time they have lost back-to-back games at the Liberty under Clement, who had never previously suffered consecutive defeats on home soil in his managerial career.