Hungarian midfielder Szoboszlai's neat finish put Arne Slot's title favourites ahead in the rain at Anfield, before Callum Wilson missed an opportunity to draw the visitors level on Merseyside.
But Mac Allister's powerful strike in the second half made sure of the points for the Reds, who opened up a 13-point gap at the top of the Premier League table - with Eddie Howe's United still just three points shy of a top four place.
The Magpies were searching for a first win at Anfield since November 1995, when a Steve Watson strike settled a League Cup tie, with this midweek contest something of a dress rehearsal for the Carabao Cup final in just over two weeks' time.
But without the services of leading goalscorer Alexander Isak, who was missing with a slight groin problem, this had all the makings of a far less mouthwatering occasion than that trip to Wembley for those in black and white - particularly once Szoboszlai swept home Luis Díaz's cutback with just 11 minutes gone.
Newcastle - who had survived a penalty appeal just before that opener when Mo Salah tumbled over a falling Lewis Hall - were still resolute and disciplined, with Hall cutting out a Diogo Jota cross before Salah could pounce on it from close range. Kostas Tsimikas sliced well wide after stepping inside off his favoured left foot, while at the other end openings were scarce. United's best one of the half fell to Wilson, who skewed his shot well wide after being played in over the top by Dan Burn.
Wilson was starting his first Premier League game of the season and led the line valiantly, if without reward. But the hosts carried a much greater threat, with Szoboszlai getting too much bend on a first-time effort which perhaps should have made it two and Salah curling just over before the break.
Howe's charges weren't quite able to turn the heat up on the hosts as much as they would have liked in the second period, despite some clumsiness in possession from the league leaders, and in truth the game was put to bed just after the hour mark. Argentina midfielder Mac Allister created the chance for himself with a driving run at the heart of the Newcastle backline. He played in Salah on the right, and the Egyptian's neat lay-off back to the former Brighton man was rewarded with a thumping finish beyond Nick Pope.
It was a little harsh on the Magpies, who had looked good after the restart. But even while not at their clinical best, the Reds proved capable of keeping them at arm's length.
Lewis Miley, Kieran Trippier, Harvey Barnes and William Osula were all sent on by head coach Howe as part of a quadruple substitution as he looked to salvage something from the final quarter, but United couldn't recover. The hosts could actually have had more, with Salah's sumptuous outside-of-the-foot cross almost creating a third for Díaz, who turned it wide at full stretch, before Salah himself then forced a parry from Pope with a low shot. They did have the ball in the net again at the death, but substitute Cody Gakpo's strike was chalked off after a foul on Pope by Ibrahima Konaté in the build-up.
The size of United's task at Wembley against the same opposition next month is clear - but before that, Howe's men will look to progress in another competition, as they welcome Brighton to St. James' Park in the Emirates FA Cup on Sunday, with a quarter-final berth up for grabs.
