Leon Osman headed the visitors into a deserved lead after 12 minutes. However, the midfielder then deflected Denilson's shot past Tim Howard as the Gunners levelled before the break.
Steven Pienaar's breakaway goal looked to have won it with eight minutes to go, only for Rosicky to grab a crucial strike in added time as Arsenal cut the deficit on leaders Chelsea down to three points.
Gunners boss Arsene Wenger had called on his team to maintain the momentum which has dragged them right back into the title rave with five wins from six but it was Everton who made the better start.
Louis Saha had got ahead of the defence on the right but blasted over.
Denilson, just returned from a back problem, had to be taken off on a stretcher after collapsing in the build up to the Everton attack. Fran Merida came on.
Arsenal threw plenty forward during the closing stages and were rewarded in stoppage time when Rosicky smashed home a close-range shot which deflected off Lucas Neill.
Wenger accepted Rosicky's late strike, which deflected in off Lucas Neill, could yet prove crucial come the end of the season. He said: ''It is an important goal because it shows the spirit of the team right until the last minute
The real tests, as we all know, will come at the end of January. Will we require new signings by then? If Denilson’s injury is long-term, then perhaps a replacement for Song will be on the agenda. On separate occasions both Gallas and Vermaelen required treatment from the physio, underlining our need for a back-up centre-back (I occasionally forget that Johan Djourou even plays for the club, so long has he been out injured). And upfront, Andrey Arshavin struggled to make an impact in a position he’s clearly not purpose-built for. It’s becoming a shame not to see Arshavin attacking from deep, where he’s at his best. Our most skillful player with the ball barely gets it.
Arsene now has a week without a fixture in which to go fishing on the market. Let’s see if he catches anything.