Fans held up shirts with Mead's name as the team's lined up, while there were chants of "Meado" ringing out around the Emirates from the first whistle.
One of the club's most pivotal players of the last decade, Mead was the liveliest player on the pitch in the opening half as she made her final home appearance for the Gunners.
But she was denied an 87th goal for the club when her header bounced down off the crossbar and was collected by Brosnan in the 16th minute.
Mead rattled the bar for a second time with a sweeping strike before the break, but she couldn't fashion any more chances and she walked off to a raucous standing ovation when she was substituted for Caitlin Foord in the 63rd minute.
"To get the response off the pitch from the fans and my team-mates, that will live with me forever. Goosebumps throughout my body. I've been proud to represent this club," Mead told Sky Sports after the game.
"Maybe I should have put a couple in the net tonight. I think I called myself a postwoman today."
Despite limited playing time this season, Mead will leave Arsenal as a legend, while her first-half performance gave any interested clubs a glance at what she can offer.
Mead thanked the fans in a post-match presentation, adding: "I've had some of my most amazing memories and also the toughest moments in my life. I lost my mum while I was here and I would never have got through that without these amazing people.
"I don't think a thank you will be enough. You guys are the best fans in the world."
She had a smile on her face as the full-time whistle blew, relieved her side could grab all three points late on - even if it was in a controversial manner.
Everton's players and staff were less impressed and a distraught Brosnan was consoled by her Republic of Ireland team-mate Katie McCabe.