James Tarkowski's second-half header sealed Everton's first victory since October, briefly moving them out of the relegation zone - although they ended the day back in it.
While there was joy on the blue side of Merseyside, Liverpool's miserable league campaign continued as they were crushed 3-0 at Wolverhampton Wanderers.
Reuters reports Liverpool trailed 2-0 after only 12 minutes with Joel Matip's own goal and a debut effort for Craig Dawson. Ruben Neves added a third for the hosts after the break.
"When you start like this you don't deserve anything in a Premier League game," Liverpool manager Juergen Klopp said.
"We have to change it immediately in the next game."
Manchester United consolidated their top-four position with a 2-1 victory at home to Crystal Palace, although it came at a cost with midfielder Casemiro shown a straight red card.
United edged above Newcastle United into third place with 42 points after Newcastle were held to a 1-1 draw at home to West Ham United that left them in fourth place in the Premier League.
Arsenal's second league loss of the season left them on 50 points, five ahead of champions Manchester City who are at fifth-placed Tottenham Hotspur on Sunday.
With no signings in the January window and Everton in a horrendous run of form, the omens were not good for Dyche as he began his rescue mission at Goodison Park against an Arsenal side whose only previous league defeat came in September.
But Everton took full advantage of a rare Arsenal off-day with Dwight McNeil's corner being powered home by Tarkowski -- two players who had worked under Dyche at Burnley.
"I've just said to them that it's a start point, that's all it is. We're laying down the fibre of what we want to be as a group but this is only a start point, no matter who you beat," Dyche said of Everton's first win in 11 in all competitions.
For Arsenal it was a wake-up call and they will be hoping for a favour from Tottenham, who play second-placed Manchester City on Sunday.
"From our side, we're disappointed because obviously we didn't get the result that we wanted, but the performance doesn't reflect what we've been doing," the Spaniard said.
Arsenal host Manchester City on February 15.
Southampton remain rooted to the bottom after a 3-0 defeat at Brentford, while Bournemouth are second from bottom after losing late on at Brighton and Hove Albion who moved into sixth spot thanks to Kaoru Mitoma's late goal in a 1-0 win.
Leicester City also allayed their relegation fears as they picked up their first league win since November, twice coming from behind to win 4-2 at Aston Villa.
Everything was going smoothly for Manchester United after Bruno Fernandes converted a penalty and then Marcus Rashford scored his 11th goal since the World Cup to make it 2-0 at Old Trafford.
But Casemiro was dismissed in the 70th minute for grabbing Will Hughes around the neck and Palace made it a nervy ending when Jeffrey Schlupp halved the deficit.
United saw out a 13th consecutive win at Old Trafford but they will now be without Casemiro for three games.