Tyler Morton and Max Woltman were also on the mark for Barry Lewtas’ charges.
Young Scouser Harvey Davies made his debut for the Reds while Lewtas was still without injured strike duo Fidel O’Rourke and Layton Stewart, and centre-back Billy Koumetio was also ruled out.
Morton fired the Reds into the lead before the Saints fought back to lead 2-1.
Woltman then levelled to make it 2-2 before Beck netted the winner deep into stoppage time.
This was Liverpool U18s final group game in the Premier League Cup and they must now wait on other results to see if they can qualify as one of the best runners-up.