Ramsey scored his eighth goal in nine matches to add to an earlier strike from teenager Serge Gnabry, and the Gunners held on after Ben Davies pulled a goal back for the hosts.
Arsene Wenger's men now have a two-point cushion over Tottenham, with Chelsea a further two points back.
That is prompting hopes that the Gunners' long wait for a trophy could be over as they put themselves firmly in the title race.
"That's our target," Ramsey told Sky Sports 1. "We have a lot of quality. We've had a lot of injuries recently but we've managed to dig in and put a load of wins together and hopefully now we get players back and we're in with a shout."
Ramsey was pleased with Arsenal's performance against Swansea, especially in the second half after a slight change of tactics.
The Welshman explained: "Swansea is a difficult place to come but we've passed the ball around really well.
"We had to dig in and stay solid and try not to concede an early goal. We've done that, we've started to defend as a unit and the goals have come in the second half.
"We knew it was going to be a tough last 20 minutes but we managed to dig in and get the win.
"In the first half we tried to press them a bit too high up but in the second half we let them have the ball, we waited for them to come into our half and defended quite deep so we could get them on the counter-attack.
"We broke with pace and quality and managed to gain a few times."
Ramsey fired home the second goal in the 62nd minute, and he is delighted with his own goalscoring run.
"I felt like throughout my career I've been getting into these positions but I was perhaps a bit rash in front of goal," he said. "I'm a lot more composed now, I'm feeling confident and things are starting to click into place quite nicely."