The tie had been switched to St. James' Park after the League Two side's pitch was severely damaged by flooding last week.
Newcastle made eight changes to the side which started Saturday's 1-1 Premier League draw with Manchester City and the Magpies struggled to hit top gear.
But Schar's spot-kick in first-half stoppage time after Miguel Almiron had been felled inside the area was enough to book their place in the last 16.
Newcastle have not won a major trophy since 1969 and the pressure is on Eddie Howe to deliver silverware for a club backed by the Saudi sovereign wealth fund.
Chelsea's trip to the northeast and Tottenham's clash with Manchester City are the standout clashes in the fourth round.
Liverpool visit Brighton, Arsenal will travel to Preston and Manchester United entertain Leicester.