A poor and scrappy game looked set to end all square when Shawcross, who was a tower of strength at the centre of Stoke's defence, rose to meet Delap's trademark delivery and bury his effort beyond a stranded Brad Jones.
The stakes were massive and it showed during an untidy opening spell, with both sides struggling to string two passes together.
Stoke did land the first blow in the seventh minute, when Jones pulled off a fine save to keep out James Beattie's close-range shot from Salif Diao's knock-down.
But Boro were holding their own and Tony McMahon squandered an excellent opportunity when he headed Andrew Taylor's pinpoint cross straight at Thomas Sorensen.
Marlon King, who scored for Hull earlier this season at the Britannia Stadium, threatened to repeat the feat before he dragged his effort wide of Sorensen's right-hand post.
The nerves were still jangling and cautions for Emanuel Pogatetz and Stoke's Ricardo Fuller came as no surprise as the free-kick count rose rapidly.
Further yellow cards for City's Glenn Whelan and Diao then infuriated a frustrated home crowd as a tedious first period moved towards its conclusion.
Stoke boss Tony Pulis replaced full-back Andy Wilkinson with Stephen Kelly at half-time, and introduced Matthew Etherington for an out-of-sorts Diao six minutes after the break.
Gareth Southgate responded by bringing on Afonso Alves for King in an attempt to pep his troops and the visitors did start to look more threatening.
Sorensen finally had to make a save of note in the 58th minute when he plunged to his left to tip around a fierce 20-yard drive from Gary O'Neil.
And Tuncay should have done better than snatch his shot poorly wide after Stewart Downing's free-kick somehow sailed through a crowded penalty area.
A poor quality contest was now at least producing some chances and Beattie fired over for Stoke before Shawcross headed a Liam Lawrence free-kick into Jones' grateful arms.
And City should have taken a 72nd-minute lead when Lawrence produced the best piece of skill in the game to outwit Taylor and hang up a cross which Beattie headed over from eight yards.
It was beginning to look as though a point each beckoned, but Delap and Shawcross had other ideas.