The Riversiders, missing Alan Coar, Josh Westwood and Jake Salisbury, won 4-3 on penalties at Swallownest on Saturday following a 1-1, extra-time draw.
It put them into the last 128 of the competition, where they will host Sunderland RCA on Saturday, November 3.
Their opponents are currently second in division one of the Northern League with victory also worth £900 to the winners.
“You’d probably say that we ground it out on Saturday,” Baker said.
“To be honest, the game could have gone either way and I said that to their bench, who were really good with us all day.
“The weather didn’t allow for pretty football as, when we left Garstang, it was throwing it down and then we got to South Yorkshire and it was bright sunshine – but with a really strong wind.
“It’s amazing how the lads are sticking together given Josh is our best centre-half and Alan is our best striker but they have gelled really well.”
The Riversiders now have a free midweek before returning to action on Saturday with a Macron Cup tie against Silsden.
Baker’s players go into the game on the back of two penalty shootout victories with the Swallownest success following their win at Longridge Town a week earlier.
Both wins were achieved in no small way by the performance of keeper Will Kitchen, who has saved a number of spot-kicks this season.
“I said on Saturday that I felt confident going to penalties because we had a keeper who is big and brave while our lads were very calm and composed,” Baker said.
“I was worried about losing the game in 90 minutes but, once we got to extra-time, I was really relaxed – I don’t know why that was, I just had that feeling.”