Neither side could break the deadlock in the second half so the game went into extra-time.
An enforced substitution was inspirational for Boro as Nathan Fisher scored with his first touch from a Jonathan Franks cross.
Boro hung on to earn a place in the fifth round, the last sixteen.
Boro team:
Steele, Honeyman, Bennett, Langthorne, Johnson, Smallwood, Franks, Robson (Fisher), Thompson, Walker (Hillerby), Porritt