Aidy White and Jose Baxter both hit the woodwork for the Blades as it ended goalless at the Lamex Stadium.
With the help of strong winds that affected the game throughout, White saw his cross-turned-shot from the edge of the area helped onto the post by Boro goalkeeper Chris Day.
Then three minutes after the break, Baxter also saw his powerful 12-yard effort cannon back off the top of the post.
The visitors dominated the game, and Day made good saves to keep out Neill Collins' header and White's swerving free-kick.
Westley, whose side remain in the relegation zone, said: "It was a very hard-earned point.
"The conditions were the winner. It was very right tough out there, a right-to-left wind, and in the first half it was really up.
"We had to defend our goal with real resilience. It was a game in which our back four were tested, and I think the players got through to half-time.
"We turned the game around, I think we had the better of the second half, although Sheffield United came strong near the end. I think it was a fair result, neither side really did enough to win it, although both could have done.
"It was difficult to get the ball down and keep it under control, so as a consequence there were a lot of back-to-front balls played.
"It was the type of game I thought would be won from a set-play. We had to make sure we didn't make any mistakes, and that we kept our concentration, work ethic, and discipline in the hope you could win it."