Phil Jagielka, who was called into the starting XI as a replacement for Michael Keane shortly before kick-off, poked home the only goal of the game following Dominic Calvert-Lewin’s header from Lucas Digne’s early long throw.
“I’m really, really happy. We deserved the three points,” Silva told BBC Sport.
“The chances we created, we should score more goals. It is a fantastic win and one more clean sheet. That’s the spirit I like – the desire to win every challenge.
“[Earlier in the season] we lost some confidence, we had a break and after we worked really well. It’s the same team with more confidence.”
Arsenal were poor in the first half and, while they did improve in the second, they never seriously tested Jordan Pickford in the Toffees goal.
Everton, meanwhile, had plenty of chances to double their lead in the second half as Bernard, Richarlison and Gylfi Sigurdsson all spurned chances.
Victory for Everton means they leapfrog Watford into ninth while Arsenal remain in fourth.
Everton only announced Jagielka was starting 15 minutes before kick-off – a late replacement for Keane who dropped out of the matchday squad due to illness.
But it only took the club captain 10 minutes to get on the scoresheet as he capitalised on confusion in the Arsenal box to fire home his first goal since April 2017 from close range.
The hosts went on to dominate for much of the first half with Brazilian pair Richarlison and Bernard causing problems as they found plenty of space in behind the Gunners’ wing-backs.
While the visitors did improve after the break with the introduction of Aaron Ramsey and Pierre Emerick-Aubameyang, the Toffees were still largely the better side.
Bernard should have doubled his side’s lead on 59 minutes when an error by Ainsley Maitland-Niles presented him with a one-on-one with Bernd Leno which the keeper saved well.
And there were more chances for the hosts as the game drew to a close, too, as Sigurdsson could only shoot straight at Leno from Richarlison’s cut-back, while the Brazilian skewed his own effort wide with the goal gaping.
There were further opportunities for Bernard, Andre Gomes and substitute Theo Walcott as the Toffees piled on the pressure but they could not find a second goal.
However, they were not made to pay for their lack of cutting edge as they claimed a victory which sees them string together three consecutive league wins for only the second time this season and the first since October.
