Palace first threatened on seven minutes when Aaron Wan-Bissaka made a surging run down the right and played a low ball across the box to Max Meyer. The Eagles’ number seven let it run on to his left foot and thumped it goalwards, but Swansea got bodies in the way.
Meyer was involved again five minutes later when he exchanged neat passes with Jeffrey Schlupp down the left before finding Jason Puncheon in space in the box. The captain took a touch to turn and flashed a powerful effort just wide of the far post as Palace began to grow into the game.
The Eagles continued to push and Sørloth had a go shortly after, but his volley was blocked behind for the first of three consecutive set-pieces, all of which Swansea coped with.
Yan Dhanda and Dan James looked the Swans’ most likely threats as both caused problems with their pace down the flanks, but neither really tested the Palace resolve.
With minutes left of the first period Sørloth had another opportunity to open the scoring, but his header went over from a Puncheon corner. That was about it for the first half’s action and the sides went in scoreless in south Wales.
Palace started the second spell strongly, but there were few clear-cut chances and it was Vicente Guaita who was called into action to deny the home side with 66 minutes on the clock. Dhanda and James combined, with the former playing the latter through with a no-look pass, but the Eagles’ stopper stood tall and parried the winger’s effort behind for a corner.
That Swansea attack sparked the game back into life, but unfortunately for them first blood was Palace’s. Jairo Riedewald received the ball in space out on the left, got his head up and played an eye-of-the-needle pass to the feet of Sørloth, who burst into the box and stabbed home the game’s opener on 70 minutes.
Palace kept Kristoffer Nordfeldt busy in the Swansea goal around the 80-minute mark, with Meyer, Sullay Kaikai and Puncheon all being denied in quick succession.
Little else happened in the final minutes and Palace held the lead to progress to the third round.
Palace: Guaita; Wan-Bissaka, Kelly (Ward, 62), Inniss, Schlupp (Townsend, 62); Meyer, Riedewald, Kouyate; Puncheon, Williams (Kaikai, 76), Sørloth. Subs not used: Henry, Woods, McGregor.
Swansea: Nordfeldt, Carroll (Asoro, 77), James, Grimes, John, Naughton, Byers, Dhanda, Baker-Richardson (Cullen, 77), Harries, Cooper (Reid, 61). Subs not used: Mulder, McKay, Maric, Lewis.