Performance in losing 1-0 to local rivals Sunderland on Sunday heaped the pressure on Eddie Howe’s men to respond.
Yoane Wissa opened his Newcastle account on his first start after just 10 minutes but Sasa Lukic quickly levelled for the visitors.
The tie appeared destined for penalties until Miley got a flick to Sandro Tonali’s enticing delivery.
Newcastle ended a 70-year wait to win a domestic trophy when they won the League Cup last season and Howe is eager for more.
“It was important to bounce back from Sunderland, to come back in a positive fashion,” he said. “We want to be competitive and win trophies – this one we had a great experience last year we want to do that again.”
Chelsea booked their place in the two-legged semi-finals early next year with a 3-1 win at Cardiff on Tuesday.
Arsenal host Palace in the final quarterfinal tie next week.
