Arsenal now has 15 points from six matches.
The victory also marked 17 years since Arsene Wenger was appointed as Arsenal’s coach.
The first 45 minutes at the Liberty Stadium was barren as chances were few and far between for both teams.
Arsene Wenger’s side eventually went ahead just before the hour mark when Ramsey released Kieran Gibbs to shoot past Swans goalkeeper, Michel Vorm.
Ramsey scored Arsenal’s second goal just minutes later after latching onto Giroud’s pass.
Swansea not willing to just go down in such a hapless manner poured themselves forward and gave themselves a lifeline with eight minutes left to play when Ben Davies cut in from the left, picked out Wilfried Bony and after picking up the striker’s flick, volleyed home past Wojciech Szczesny.
However, that goal proved to be too little too late as Arsenal saw out the remaining minutes to pick up the three points that has seen them pull off the crowd and now enjoy a two-point gap at the summit of the EPL table.
Arsenal now has 15 points from 6 games, two points ahead of Tottenham who drew 1-1 against Chelsea.