Holgate headed home what was his first goal for the Toffees in the 72nd minute, and ex-Hornets man Richarlison fired past Heurelho Gomes in stoppage time.
After a first half low on entertainment, the Merseysiders went close three times before breaking the deadlock, with Andre Gomes having a shot cleared off the line and efforts from Alex Iwobi and Lucas Digne hitting the woodwork.
Ten minutes later, Everton were bursting forward again, but Richarlison messed up his attempt to pick out a team-mate, and Domingos Quina then skewed wide at the other end.
Iwobi then struck an effort, following a mistake by Chalobah, that was comfortably caught by Heurelho Gomes as a lacklustre first half neared its conclusion.
Silva brought Theo Walcott on for Kean at the break and the substitute almost combined with Iwobi in the opening moments of the second half, being beaten to the ball by Heurelho Gomes.
Richarlison sent a curling effort off-target, before a fierce Andre Gray shot brought a decent save out of Jordan Pickford.
Keane saw his bouncing header dealt with by Heurelho Gomes in the 61st minute, and Andre Gomes then saw his shot go past the goalkeeper, only for Adrian Mariappa to clear off the line.
Soon after Silva's men hit the woodwork twice in quick succession, Iwobi's strike hitting the corner of the goalframe and Digne's free-kick then deflecting against the bar.
And the increased pressure then paid off moments later as Holgate headed Everton in front, converting from Walcott's cross.
Richarlison wrapped things up at the end as he burst into the box and cracked an emphatic effort beyond Heurelho Gomes.