The FTSE 250 group said its Ocean Guardian rig, which left Scotland at the end of November, arrived on Friday and the Liz 14/19-A exploration well was spudded yesterday afternoon.
The well is being drilled to a target depth of around 3,500 metres with drilling operations expected to take around 30 days.
This marks the start of an extensive drilling programme for the AIM-listed company in the region.
The well on the North Falkland Basin will be the first time the area has been drilled since oil major Shell (RDSB) abandoned its efforts in 1998.
The drilling programme became the subject of much speculation last week after Argentinian President, Cristina Kirchner, declared that all ships making their way from its ports to the Falkland Islands, South Georgia or the South Sandwich islands require a government permit.
She accused Britain of refusing to comply with international law by allowing British oil explorers to embark on drilling activities in the region.
This escalated the diplomatic row between Argentina and Britain over ownership of oil beds situated within the Falklands' waters following Argentina's claims that British oil companies are exploiting its reservoirs.
The Falkland Islands government says: "The Falkland Islands government has every right to develop a hydrocarbons industry within our waters. The British government has clearly stated that they support our right to develop legitimate business.
The British government have also reiterated their stance on our British sovereignty.
"It is no surprise to anyone that Argentina is behaving in this way but it is nonetheless disappointing when they do."
UK defence minister Bill Rammell has vowed to take whatever steps are necessary to protect the islands.
The Falklands is also home to Rockhopper Exploration (RKH), Falkland Oil & Gas (FOGL), Borders & Southern Petroleum (BOR) and, most recently, mining heavyweight BHP Billiton (BLT).