The visitors, who started the campaign with a six-point deduction after a match-fixing investigation, picked up their second win of the season as Denis struck either side of half-time.
Djamel Mesbah scored a controversial first-half goal for Lecce, who had Carlos Grossmuller sent off shortly after Atalanta's winner.
Denis opened the scoring in only the third minute, slotting home from the penalty spot after Mesbah brought down Ezequiel Schelotto.
Lecce responded strongly in the remainder of the first half, though, and Christian Pasquato shot just over and then scuffed another effort wide after being teed up by David Di Michele.
Grossmuller's 20-yard drive was comfortably fielded by keeper Andrea Consiglie before the hosts scored a deserved equaliser.
The strike itself was disputed, though, with Mesbah seemingly in an offside position from Pasquato's touch before poking the ball home from close range.
Consiglie was then forced off by injury and replaced by Giorgio Frezzolini, but Atalanta regrouped and threatened twice before half-time.
Giacomo Bonaventura shot high and wide before a more dangerous effort from Maxi Moralez flashed narrowly wide.
Denis curled a free-kick over the bar early in the second period, but scored his second in the 56th minute when he half-volleyed home after Daniele Capelli nodded a corner down into his path.
Grossmuller was dismissed in the 63rd minute when he was booked for a bad foul on Moralez and then received a red card after taking his complaints to the referee too far.
Di Michele shot wide from distance and Denis immediately responded in kind.
Atalanta had two further chances to extend their lead, Giampaolo Bellini and Denis both shooting wide, but the Argentinian's double proved sufficient.