Virgil van Dijk repaid part of his record €75 million transfer from Southampton with a late header against Everton to send Liverpool into the fourth round of the FA Cup.
Milner put Liverpool in front from the penalty spot ten minutes before the end of the first half.
Everton drew level after the break when Sigurdsson struck from the edge of the area.
A Merseyside derby that had plenty of passion but never boiled over looked to be heading for a replay at Goodison until Van Djik on his debut headed past Pickford to send the Kop and Anfield into raptures of delight.
In the night’s other FA Cup tie, Manchester United left it late to see off a stubborn Championship side Derby at Old Trafford with Lingard and Lukaku scoring inside the last ten minutes.