Vale, who played the majority of the match with 10 men after Daniel Jones' 20th-minute red card, had threatened to spoil the party when Adam Yates' fluked equaliser 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.
Collins headed United in front from Stephen McGinn's corner before Jones' scything challenge on Ryan Flynn brought a red card for the defender.
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.
Lyle Taylor, having earlier been inches away from hooking home a close-range effort, drew a superb reaction save from Vale goalkeeper Chris Neal before seeing another impudent flick deflected wide.
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.