[Minor edits 12/26/09]
The American victory at Cowpens is often credited to a double envelopment of the British forces (see this map on John Moncure's The Cowpens Staff Ride and Battlefield Tour webpage).
This is not quite how the final stages of the battle are portrayed in some accounts. In addition to the pressure against their outer flanks, there was also a collapse of the middle of the British line.
Lieutenant-Colonel John Eager Howard, who commanded the Continental infantry, wrote that at the climax of the battle: "The enemy were now very near us. Our men commenced a very destructive fire, which they little expected, and a few rounds occasioned great disorder in their ranks. While in this confusion, I ordered a charge with the bayonet, which order was obeyed with great alacrity."
Howard's Continentals charged into the center of the British infantry, breaking their line into two pieces.
Sergeant William Seymour of the Delaware Continentals wrote that when this gap was created, the Continentals advanced and fell upen the flanks of the British to their left and right: "..Captain Kirkwood with his company wheeled to the right and attacked their left flank so vigorously that they were soon repulsed, our men advancing on them so very rapidly that they soon gave way. Our left flank advanced at the same time and repulsed their right flank, upon which they retreated off, leaving us entire masters of the field."
Along with Kirkwood's company, Howard participated in the attack towards the right: “In the pursuit, I was led towards the right, in among the 71 st..."
Major Joseph McJunkin of South Carolina also described these two halves. “One battalion [actually one half of the British line] throws down their arms and the men fall to the earth. Another commences flight, but Washington darts before them with his cavalry and they too ground their arms."
David Stewart's (1825) Sketches... of the Highlanders of Scotland indicated that this break in the center of the line was the primary reason why the 71st Foot, on the left of the British line, was forced to retreat during the battle:
"They [i.e., the 71st] did not immediately fall back, but engaged in some irregular firing, when the American line pushed forward to the right flank of the Highlanders, [the 71st] now realized that there was no prospect of support, and while their number was diminishing that of their foe was increasing. They first wavered, then began to retire, and finally to run. This is said to have been the first instance of a Highland regiment running from an enemy. This repulse struck a panic into those whom they left in the rear [i.e., Tarleton's reserve of British Legion dragoons], and who fled in the greatest confusion. Order and command were lost, and the rout became general. Few of the infantry escaped, and the cavalry saved itself by putting their horses to full speed."
Loyalist Alexander Chesney did not specifically mention the collapse in the center of the British line, but he did confirm that at this point in the battle, the 71st was "unsupported."
It is unclear which British regiment took the brunt of Howard's counterattack. As previously mentioned, Morgan and Tarleton offered different descriptions of how the British units were arrayed. According to Morgan, it was the infantry of the British Legion that were most directly opposed to Howard's Continentals. According to Tarleton, it was the 7th Foot.
Tarleton's description of the battle can be found on Marg Baskin's Banastre Tarleton website.
David Stewart (1825). Sketches... of the Highlanders of Scotland.
Morgan's report of the battle can be found in James Graham's (1856) The Life of General Daniel Morgan.
John Moncure's online history of the battle, The Cowpens Staff Ride and Battlefield Tour, includes a transcription of statements by Morgan, Tarleton, Howard, McJunkin, Chesney, and Seymour.
For McJunkin's recollections, see James Saye's Memoirs of Major Joseph McJunkin, Revolutionary Patriot. Also see this issue (.pdf file) of the online newsletter, Southern Campaigns of the American Revolution.
A transcription of William Seymour's journal can also be found on this Battle of Camden website.