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 handed a debut to deadline-day signing Ozil, while Santi Cazorla missed the trip to Wearside. Per Mertesacker was also absent through illness, so Bacary Sagna moved to centre back.
Paolo Di Canio handed a debut to Ki Sung-Yueng after his loan move from Swansea City, while Steven Fletcher started a game for the first time in six months after scoring on his return from injury as a substitute at Crystal Palace before the international break.
Arsenal dominated possession in the opening exchanges and it took just 10 minutes for Ozil to make an impact on the Premier League.
The Germany international latched on to a punt from Kieran Gibbs down the left flank. He showed a neat first touch before squaring for Giroud to calmly slot home at the near post.
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.
And Walcott made it a hat-trick of missed opportunities three minutes before half-time. Jack Wilshere surged through and passed to Walcott, whose first touch saw the ball flick into the air and he headed the effort wide.
Sunderland needed a quick start and drew level three minutes after the restart.
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.
It was Arsenal, though, who finally grabbed the goal their play warranted in the 68th minute.
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.
However, the referee had already blown for a free-kick in favour of the Sunderland forward and the goal was ruled out.
And their sense of misfortune was confirmed with 15 minutes remaining when Ramsey grabbed his second of the game.
A neat move saw Ozil find Giroud as he flicked a first-time pass to the Welshman, who calmly slotted beyond Westwood.
And Di Canio's afternoon got worse in injury time when he was sent to the stands for arguing with the fourth official.
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