Anthony Elanga's early goal had given the Magpies a 1-0 half-time lead on Tyneside amid a raucous atmosphere.
Cody Gakpo briefly levelled the scores from range at the start of the second half but substitute Joe Willock's superb finish - which came after a brilliant run from Yoane Wissa - just a couple of minutes after entering the fray swiftly put the hosts back in front.
Despite some late Reds pressure it looked like being the winner until the very last seconds of added time, when Lewis Hall brought down Víctor Muñoz in the box. Referee Stuart Attwell pointed to the spot, allowing Szoboszlai to blast home the equaliser and deny Matthias Jaissle a win in his first competitive game in charge of United.
After the moving pre-match tributes to Kevin Keegan - an iconic figure at both clubs - Lewis Hall blocked an early Florian Wirtz effort but barely four minutes in, United broke at pace, with Osula feeding Elanga. The Swedish winger's low finish was unerring, beating Alisson in the Reds' goal and nestling in the bottom right-hand corner to give Jaissle and his side a stunning start in front of their own supporters.
Lukáš Horníček - one of three debutants in the Newcastle line-up, along with Amar Dedić and Sean Steur - got down well to deny Wirtz just after the 15-minute mark before palming away a dangerous Milos Kerkez cross as the visitors began to probe.
But the St. James' Park faithful were responding to their team's wholehearted efforts and Osula's pace in behind gave him his first sight of goal, with Alisson smothering the shot before the rebound just eluded the United frontman.
Horníček was called into action straight after the restart and Liverpool tried to force home a leveller from close range following a corner, but United were a constant threat on the break, and that chance for the Reds led to another counter-attack. Wissa, lively in the number ten role, forced Osula just too wide with his through ball but after possession was recycled, Lewis Miley's well-struck drive flashed just over the crossbar.
Gakpo fired just off target soon after and it served as a warning, because nine minutes into the second period the Dutch forward levelled the scores. It was a fine low shot from Gakpo, which skidded past Horníček and into the bottom left-hand corner from a position the Magpies may have felt he was more likely to pass from.
But just after his introduction, Willock steered Newcastle back into the lead following some excellent work from the tireless Wissa who drove at the heart of the Liverpool backline before feeding the substitute, who made no mistake from just outside the box.
Wissa, enjoying perhaps his best afternoon in black and white, narrowly failed to connect meaningfully with a Hall cross before Elanga reminded Alisson of his side's threat in the final third with a curling strike that the Brazilian stopper held.
And with five minutes left, the former Brentford man had a glorious chance to make it 3-1 and put the game to bed when Willock's hooked ball over the top released him, but Alisson again rushed off his line to save with his legs.
With five minutes of added time already played, they were made to rue that miss when Hall upended Muñoz in the penalty area. Szoboszlai stepped up to smash a scarcely-merited leveller past Horníček and snatch a point for the visitors with almost the final kick of a thoroughly entertaining encounter between the two clubs that Keegan held dearest.
Newcastle United: Lukáš Horníček, Lewis Hall, Sven Botman, Yoane Wissa (Fabian Schär 90), Will Osula (Jacob Ramsey 56), Harvey Barnes, Malick Thiaw, Sean Steur (Joe Willock 56), Anthony Elanga (Jacob Murphy 75) Amar Dedić, Lewis Miley (Aladji Bamba 75)
Subs not used: Ewen Jaouen, Bazoumana Touré, Nick Woltemade, Leo Shahar
