The hosts led twice through goals from Scott Laird and Craig Reid but were pegged back by Steven Schumacher and Nicky Ajose.
When Schumacher scored again to put the Shakers ahead it looked bleak for Stevenage, but substitute Ben May levelled just three minutes later.
Westley said: "Well I think that game had everyone on the edge of their seats - It had everything. Of course it's a major relief that we've managed to sneak into the play-off positions, but overall I think the lads have worked so very hard all season and we deserve to be there.
"We knew full well it would be a very tough challenge for us, even though some people may have felt otherwise as Bury had already secured their promotion.
"We prepared hard during the week and we got ourselves off to a flyer early in the game. On another day we could have scored six or seven, but at least we've created chances and looked lively all day. That bodes well for what awaits us in the play-offs now."
Source: PA
Source: PA