Boro dispensed with the lowest ranked team in the competition's second round thanks to goals from Francois Zoko and Lucas Akins either side of half-time at Broadhall Way.
Luke Freeman added a third on the hour and Filipe Morais slotted a late penalty as Stevenage claimed their first win in three in all competitions to progress to the third round.
But Boro boss Westley admitted it wouldn't have come as a surprise to him had non-league Stourbridge caused an upset and turned the tables on Stevenage.
"We've been on a bad run of form with a couple of indifferent performances and difficult results and the performance wasn't really a lot better," said Westley.
"It was one of those that we had to dig deep in, grind it out and find enough quality to win through in the end. I think the side looked intimidated by the occasion.
"There were a lot of lads who were feeling the pressure and struggling to deal with it. Not everybody is used to winning trophies, winning medals and making decent things happen.
"When you put that pressure on them, not everybody responds in the right way and I think we have seen a lot of the players not responding in the right way to the pressure of having to win.
"That's part of their learning curve. They've got to understand that we're not here to mess about; we're here to put trophies on the table. We're not here to make excuses and come second."