After Clayton Donaldson had cancelled out James Hayter's opener, Stech denied Paul Hayes from the penalty spot 20 minutes from time.
Hayter put the visitors back in front 10 minutes later before a Tony Craig own goal confirmed victory for the Glovers.
Johnson said: "I think the penalty save was the turning point, but we showed great resilience throughout and real composure at the end to pick our counter-attacks.
"We knew pace would be the key and we have it in abundance.
"It's a good win here because Brentford are looking to be one of the sides up there at the end of the season so to come and finish so clinically was superb - that's what you get from James Hayter."