Three successive defeats had led to increased scrutiny on Silva, with Evertonhaving had to wait 17 days to try and bounce back from the loss at Watford, but former Swansea City star Sigurdsson stepped up to the mark.
Having opened the scoring four minutes before half-time, Sigurdsson curled in a superb second to ensure Everton's dismal run came to an end – becoming the Toffees' leading scorer in the process.
Things could have been different if not for some excellent defending from Lucas Digne early on, but Cardiff offered little threat as they slumped to a second straight defeat, their misery compounded by a late third from Dominic Calvert-Lewin.
Phil Jagielka almost cost Everton with an untimely slip two minutes into his first start since opening day, but Digne beat Nathaniel Mendez-Laing to Kenneth Zohore's subsequent cross.
Jagielka was involved in Everton's first chance – Theo Walcott slicing his volley wide from the defender's knockdown.
Sigurdsson made no such mistake from Everton's next opening though, side-footing home with a composed first-time finish from Seamus Coleman's cutback.
And Sigurdsson came up with the goods again in the 66th minute, drilling in on the rebound after Bernard's low cross was blocked.
Further gloss was added in the final minute of stoppage time, Calvert-Lewin capping his 100th appearance in league football with a well-taken goal to get Everton back on track in emphatic fashion.
Everton out of the woods
Everton have had a torrid run since the first Merseyside derby of the season, but Silva will now be hoping his side have put any lingering doubts over getting dragged into a relegation scrap to bed as they prepare to face Liverpool once again in their next outing.
While the Toffees can still harbour hopes of a push for Europe, Cardiff are firmly in the mire, and will slip into the bottom three should Southampton avoid defeat against Fulham on Wednesday.
Richarlison missing his spark
Having had a fine start to the campaign, Richarlison has faded in recent outings, and a lacklustre display – which was cut short after an hour – was made worse when his replacement Bernard played a crucial role in Everton's second.
