The League One side, who are bottom, face Everton for the second time this season on Saturday in what is the first meeting between the two clubs at the Lamex Stadium.
When the teams clashed in the League Cup at the end of August, Everton ran out 2-1 winners after extra time at Goodison Park thanks to goals from Gerard Deulofeu and Marouane Fellaini.
Everton manager Roberto Martinez lifted the FA Cup in May with former club Wigan Athletic and will have designs on a prolonged run in the competition this season, providing the Premier League side can sidestep a fourth-round upset this weekend.
Stevenage have lost just three of their last 11 home games in all competitions to give Westley belief that his players can produce a shock.
"We don't expect victory to come easily," he said.
"But nobody in football or associated with the FA Cup would want us to go into the game with the mentality of doing anything but win the game.
"We're under no illusions of the importance of the game and we will sweat blood to win it.
"Our very best chance of winning this tie is at the weekend, it's a difficult surface out there, it's not easy and there's more bad weather forecast, but the pitch will be playable.
"The bonus for us is that we know the surface, we know the pitch and we're geared for playing on it."
Stevenage are 61 league places below Everton and Martinez's side head into the game having only lost once in their last 16 games in all competitions.
That unbeaten run included a 4-0 victory over QPR in the third round and a 1-1 draw at West Brom on Monday.
However, Martinez is without a number of first-team regulars for the Stevenage clash.
"The only difference from the squad on Monday is Seamus Coleman," Martinez said. "He picked up a minor soft tissue injury, so he'll miss the weekend and we'll assess him from there.
"He adds to Ross Barkley and Steven Pienaar who are still unavailable, on top of the long-term injuries to Arouna Kone, Darron Gibson and Gerard Deulofeu.
"Antolin Alcaraz is still recovering well but will be missing at the weekend. Apart from that, everyone else is available."
