cancelled out Neill Collins' opener for the hosts.
The Valiants endured a nightmare start as they fell a goal behind and had Jones dismissed for a second bookable offence during the opening exchanges of the game.
Tempers continued to fray as Vale manager Micky Adams, a former Blade returning to Bramall Lane, and Jennison Myrie-Williams were involved in a furious touchline row when the forward was substituted during the ensuing reshuffle.
But, on the stroke of half-time, the visitors levelled in fortuitous circumstances when Yates' cross crept in at the far post.
Adams' side emerged as a far more dangeous proposition following the interval, despite seeing both Flynn and the impressive Taylor go close at the other end of the pitch.
United striker Shaun Miller made his long-awaited return from injury midway through the second period.
Tony McMahon and Matt Hill both flashed shots wide as the hosts looked to regain the lead but it was Doyle, powering the ball beyond Neal's outstretched arm, who obliged after Flynn and McMahon had exchanged passes.