Two goals from Merouane Zemmama and one each for Andy Halliday and Ishmael Miller saw Boro into round four, but Hastings teenager Bradley Goldberg struck a superb consolation while midfielder Lee Carey hit a post.
The Ryman Premier Division strugglers, ranked 131 places below their opponents in the league structure, trailed only to Zemmama's opener at the interval after keeper Liam O'Brien saved an early Miller penalty.
Mowbray said: "It was a credit to them that they tried to play football right from the off. I was pleased for them that they scored a great goal in front of their supporters. It was nice to be able to congratulate them afterwards.
"It was a bit patchy from us but FA Cups are about going through to the next round. It's important to keep the confidence and belief going no matter the name of the team you're playing. We got the job done, but we could have made it easier."
- Sports & Recreation
- Merouane Zemmama
- Andy Halliday
- Ishmael Miller