The best moment of a quiet first half came on 37 minutes, when Aaron Ramsdale spectacularly saved Pierre-Emerick Aubameyang's long-range shot.
The introduction of Pepe as a 58th-minute substitute provided the spark that Arsenal needed.
He released Hector Bellerin down the right and the defender crossed for Saka, who celebrated his England call-up this week with a headed goal on 61 minutes.
Less than three minutes later Pepe scored himself with a superb solo run and finish to cap off a 19-pass move.
Sheff Utd were given hope when David McGoldrick curled in their first goal of the season with six minutes left, but Arsenal held on for a win that lifts them up to fourth on nine points.
The Blades remain without a point and are second from bottom.
Arsenal: 17 Oct v Man City (A)
Sheff Utd: 17 Oct v Fulham (H)