Edmund finished a breakthrough 2016 as the world number 45.
But the South Africa-born 21-year-old was pushed hard in a 7-6 (7-4) 7-6 (8-6) win over Escobedo, whom Edmund also beat at the US Open in September.
Escobedo had a set point in the second tie-break, but Edmund set up a match with Lucas Pouille or Gilles Simon.
British number two Kyle Edmund beat American Ernesto Escobedo in two tie-breaks to reach the second round of the Brisbane International.