Ramsey scored the winner in the 12th minute following a goalmouth scramble, though Arsenal had to contend with some intense Burnley pressure to earn their win.
George Boyd missed a glorious chance to equalise and Danny Ings also came close for Burnley – but Arsenal managed to record an eighth successive victory in the Premier League to stay within sight of leaders Chelsea.
And Ramsey expressed his happiness at overcoming the Burnley threat, which has won Sean Dyche’s men plenty of admirers this season.
“It was definitely one of the tougher games,” Ramsey, who made his 150th appearance Premier League appearance, told Sky Sports. “A lot of the big teams have dropped points here this season.
“It proved another difficult game and another test that we got through. We were pretty solid and dealt with the threat – they like to get forward.
“Maybe in the past we’ve been guilty of trying to go forward and being punished. I thought we controlled the game pretty well.
“We were under pressure but we felt pretty comfortable defending and it was another win, and another win away from home.”