The defeat, their first in seven league games, was no less than deserved after a timid display that was punished when an unmarked Jagielka, in as a late replacement for the unwell Michael Keane, seized on a loose ball in the 10th minute, the 36-year-old tucking home as Arsenal's defenders appealed in vain for offside.
The arrival of Aaron Ramsey at half-time led to an improved second-half showing but the Toffees capably dealt with rare defensive concerns in rising to ninth.
Jagielka learned of his elevation to the XI shortly before kick-off and was soon celebrating his first goal in two years, Lucas Digne's long throw having broken kindly to him in the box.
That extended a shaky Arsenal's season-long wait for an away clean sheet and their defensive stocks took a hit when Sokratis Papastathopoulos earned a booking that rules him out of games against Watford and Crystal Palace.
Ramsey, who battled a groin issue during the week, stepped off the bench at the interval and quickly increased Arsenal's lean shot count.
Bernard, Gylfi Sigurdsson and Richarlison all fluffed good chances to double the home side's advantage as the second half wore on, the latter especially culpable for a wayward finish with time to pick his spot.
The misses ultimately mattered little, though, as Marco Silva's men held firm for a fifth clean sheet in six and third consecutive victory.
What does it mean? Away days an ongoing annoyance for Arsenal
Sub-par performances away from home became a theme in the latter stages of Arsene Wenger's long reign and new Gunners boss Emery has had little success in rectifying the situation.
His team have now won just once in their past nine Premier League road trips, a record they will come to rue if Champions League qualification eludes them.
1 - Arsenal attempted just one shot in the first half - their fewest in the first half of a Premier League game since May 2016 against Man City (also one shot). Blunt.— OptaJoe (@OptaJoe) April 7, 2019
Calvert-Lewin a maturing leader of the line
Goals may still be rare for Calvert-Lewin but the 22-year-old's impact in general play continues to improve. He never let Arsenal's defence relax and played a key role in Jagielka's winner.
Elneny fails audition in an awkward midfield axis
With star man Ramsey deemed only fit enough for a place on the bench, Mohamed Elneny was drafted into Arsenal's starting XI. He might not get many more opportunities, though.
The Egypt international lost possession eight times in his 45 minutes on the pitch, failed to register a tackle and, alongside Matteo Guendouzi, was part of a midfield pairing that set the tone for a meek display.
Key Opta Facts
- Everton have won three consecutive Premier League matches without conceding for the first time since January 2017.
- Aged 36 years and 233 days, Everton defender Jagielka became the oldest player to score a Premier League goal this season.
- Arsenal are one of only two teams in the top four tiers of English league football this season without an away clean sheet this season, along with Championship side Ipswich Town.
- Everton have won their last two Premier League games against ‘big-six’ opponents – as many as in their previous 41 league games against them (W2 D14 L25).
Arsenal entertain Napoli in the first leg of their Europa League quarter-final tie on Thursday before a trip to Watford the following Monday. Everton will look ahead to next Saturday's trip to already-relegated Fulham with some relish.