Ramsey set up Serge Gnabry's first goal for the club before scoring the second himself, as Arsenal issued a statement of intent to leave Arsène Wenger celebrating the 17th anniversary of his reign on top of the pile.
"Winning is better than any cake," Wenger said. "The other results came as a surprise but added pressure because you feel guilty if you don't take an opportunity. During the season you have to grab the chance when you have it.
"Swansea were sharper than us in the first half but then we played in a more compact way and took advantage every time we won the ball to be dangerous.We have seen a different Arsenal, much more clinical and playing with much more purpose."
That was embodied by Ramsey, who provided the impetus for the surge in tempo that caught Swansea cold and earned his eighth goal in as many games.
Wenger added: "People became a bit impatient with him and put pressure on him when he had some bad games. When he was questioned, that was the turning point.
"It's an important period in the life of a football player and can he deal with that? Does he go back to basics and practice even harder or feel sorry for himself and give up? When I saw him respond, that's when I thought he would always come back. It's why I kept faith in him."
Swansea stifled the visitors for close to an hour but Michael Laudrup was made to pay for a conservative team selection by two goals in the space of five minutes in the second half.
That saw Arsenal clinch a club record eighth successive away win in the League, while Swansea have now failed to win their last eight League games at the Liberty Stadium.
Laudrup said: "It has been a tough start at home but we can't change that. We have the same number of points as at this stage last season. But we want to get back to winning ways at home."
Ramsey was always assured of a frosty reception this side of Wales given his affiliation with bitter rivals Cardiff City, where he began his career. He was in the thick of it, however, nipping at the heels of his opponents which eventually brought a warning from Mark Clattenburg.
An early effort was wide but for all of Arsenal's intricacies it was as close as they got in the first half. Denied the incisiveness of Theo Walcott, handing a second League start for Gnabry, and the ingenuity of Santi Cazorla, Arsenal lacked penetration. Even Mesut Ozil struggled to dig out a moment of inspiration from the shadow of Jose Canas.
Yet even after 17 years in charge, Wenger proved he can still inspire. Whatever was said in the dressing room worked wonders. Arsenal emerged with far greater purpose and speed, they were first to the ball, quick to attack and when the goal finally came, it was sumptuous.
Ozil proved the instigator and Olivier Giroud the pivot, turning Swansea's defence inside-out before Ramsey released 18-year-old Gnabry, who finished with a confidence that defied his age.
Arsenal released the shackles and Ramsey handed Ozil what appeared a formality from close range only to be denied superbly by Michel Vorm. But when Giroud found Ramsey loitering with intent on the edge of the box, the Welshman thumped home an emphatic response to the Swansea hecklers.
Swansea finally responded when Ben Davies caught Bacary Sagna off guard to finish Wilfried Bony's neat lob but Arsenal were not to be denied.
Swansea (4-3-3): Vorm; Tiendalli, Chico, Amat, B Davies; Canas, Shelvey (Bony, 61), de Guzman; Dyer, Routledge (Pozuelo, 64), Michu.
Arsenal (4-2-3-1): Szczesny; Sagna, Mertesacker, Koscielny, Gibbs; Flamini, Ramsey; Gnabry (Arteta, 78), Ozil, Wilshere (Jenkinson, 90); Giroud.
Referee: Mark Clattenburg.
Man of the match: Ramsey (Arsenal)
Match rating: 6/10