Premier League: Burnley 0 Arsenal 1
Arsene Wenger's men are the first team to compile such a prolific run in the English top flight this season and they move to within four points of Chelsea ahead of the leaders' London derby at QPR on Sunday.
In a game of few chances, Ramsey struck decisively early in the first half after the lively Alexis Sanchez spurned a fine opportunity to open the scoring.
Burnley’s sustained attempts to get back on terms failed to bear clear openings as a lack of cutting edge in the final third once again undermined their gutsy fight for Premier League survival.
George Boyd's strike in the 1-0 win over reigning champions Manchester City last month is the only goal Sean Dyche's team have scored in their past six outings and they lie second bottom, two points from safety, with six games remaining.
Two unchanged teams provided a fluent start to the match and Arsenal goalkeeper David Ospina was out sharply to thwart Sam Vokes after the Burnley striker outpaced Per Mertesacker down the left channel.
There was a fortunate escape for Burnley when Sanchez blasted over from David Jones' loose interception, but they were unable to stop Arsenal taking a 12th-minute lead.
Goalkeeper Tom Heaton saved well with his legs to deny Mesut Ozil and Kieran Trippier blocked Sanchez's follow up before Ramsey marked his 150th Premier League appearance by lashing into the top corner.
Set-piece specialists Trippier and Santi Cazorla came close with free-kicks for their respective sides as the first half ticked past its midway point.
Goalmouth action faded towards half-time as Arsenal dominated possession and frustration began to creep into Burnley's play.
The hosts began the second period with renewed vigour and Ospina was well-placed to stop Ashley Barnes' shot on the turn when Michael Duff knocked down a 50th-minute corner.
Heaton kept out a stinging low drive from Sanchez, while Trippier briefly feared he would be culpable for an own goal when Hector Bellerin's cross struck him on the shins and trickled behind.
Boyd had another chance to be Burnley's hero against a Premier League high-flyer but missed his kick at the back post in the 70th minute after full-back Ben Mee had bustled down the left flank.
Top scorer Danny Ings then failed to make clean contact with an acrobatic effort as Turf Moor roared their team towards a rousing finale that they were unable to provide.
Substitute Danny Welbeck was unable to give Arsenal breathing space, shooting wide on the angle as Sanchez's pass proved a little too heavy, but they head into next weekend's FA Cup semi-final against Reading in high spirits.