Steve Bruce rung ten changes for the cup tie - with youngsters Reyes Cleary and Zac Ashworth being handed their first full starts.
And the former had a great chance to give Albion a first half lead but was denied by Sheffield United keeper Wes Foderingham.
Albion showed quality throughout the tight cup tie - and they took the lead 17 minutes from time when Grant headed home Callum Robinson's cross before the substitute hobbled off injured.
Robinson and then Phillips hit the bar soon after - with Albion holding out to set up a second round tie with Derby County.
The pair chatted about changes and players who impressed, Dike's injury update and links to Keinan Davis.