The Welshman took centre stage after a Craig Gardner penalty had cancelled out Olivier Giroud's first-half opener.
Frenchman Giroud became only the eighth player to score in the opening four games of a Premier League campaign, and his goal at the Stadium of Light is the first time he has notched outside of London for Arsenal.
Mesut Ozil was particularly impressive on his debut following his move from Real Madrid and provided the assist for Giroud as Arsenal led at half-time before Gardner's leveller in the 48th minute.
But Arsene Wenger's side secured the points when Ramey volleyed home from Carl Jenkinson's centre before he added his second with 15 minutes remaining.
Arsenal's celebrations were almost curtailed instantaneously as Modibo Diakite got above Giroud to crash a header against the crossbar from a corner.
But Arsenal continued to look livelier and Ozil and Theo Walcott twice combined well down the right, but the England international was thwarted on both occasions by smart saves from Kieren Westwood.
Adam Johnson plucked the ball out of the air and drove at Laurent Koscielny who slid through the winger leaving referee Martin Atkinson no choice but to award the penalty. Gardner – who had only just come on as a substitute – dispatched the ball hard and low into the right corner.
And the home side thought they had the lead in the 59th minute when Fletcher tapped home from Johnson's drilled cross but he was denied by the offside flag.
Jenkinson's cross from the right was met by a controlled volley from Ramsey who fired into the far corner.
There was a huge flash point shortly afterwards when Jozy Altidore broke free of Sagna's challenge and squirmed an effort under Wojciech Szczesny with goal line technology confirming the ball had crossed the line.
And Di Canio's afternoon got worse in injury time when he was sent to the stands for arguing with the fourth official.
