Boro had striker Lucas Akins sent off in the 80th minute for punching Saddlers defender Paul Downing after the ball had gone, referee Andy Madley producing the red card after consulting with his fourth official.
Stevenage went close in the fifth minute when a deflected drive from Jimmy Smith looked certain to fly inside the far post but Richard O'Donnell dived to his left to pull off a fine save.
Walsall cashed in on that escape by taking the lead six minutes later, Ashley Hemmings holding off a challenge from David Gray before cutting the ball back into the goalmouth where Sawyers firmly side-footed the ball home for his second goal of the season.
Eleven minutes into the second half it was two-nil as James Baxendale released Westcarr, who hammered a low left-footed shot past Chris Day for his sixth of the season.
Francois Zoko pulled one back for Boro in the 72nd minute when O'Donnell dropped a cross from Luke Freeman but it was not enough as the visitors finished with 10 men.