The 31-year-old winger was restored to the starting line-up for the last-gasp 1-0 defeat to his former club Tottenham last week after cameo appearances off the bench in the preceding fixtures against Liverpool and Brighton.
But Lennon, capped 21 times by England, sustained cartilage damage at Wembley to join the Clarets' lengthy injury list.
Manager Sean Dyche gave no indication of when he expects Lennon to return - meaning he could miss Burnley's hectic festive schedule - but stressed both player and doctor were satisfied by the outcome of Wednesday's surgery.
Dyche said: "He's had an operation, another one unfortunately. We keep getting close to getting everyone back and then a few slip away.
"He's had an operation, simple in the way it is, a cartilage operation, a sort of tidy up situation - there's no timescale on that at the moment.
"But he's happy with it and the specialist was happy with it."
Lennon is likely to be joined on the sidelines by fellow midfielder Steven Defour for their trip to Emirates Stadium in the Saturday lunchtime kick-off because of a calf complaint.
Robbie Brady also suffered a minor injury against Spurs but he and another wideman in Johann Berg Gudmundsson could come into contention as Burnley look to bounce back against the Gunners.
"There's still some way to go before we get everyone fit," Dyche added.