Daniels scored the only goal after 65 minutes at Mbombela Stadium in northeastern city Nelspruit, heading powerfully into the net off a corner.
Jele spurned a chance to equalise 11 minutes later as his timid penalty was pushed away by goalkeeper Ronwen Williams and the onrushing Justin Shonga blazed the loose ball wide.
SuperSport rose two places to fourth as the season enters the final 10 rounds while Pirates remained third.
Leaders Bidvest Wits have 37 points, defending champions Mamelodi Sundowns 35, Pirates 33, SuperSport 32 and Cape Town City 30 as the league shapes up for thrilling climax.
It was the third Premiership win in a row for three-time champions SuperSport and Daniels was thrilled after receiving the man-of-the-match award.
“We played with intensity and pride. We fought for every ball. We performed like the SuperSport of old,” he said.
“Our results in the first half of the season have been disappointing, largely because of a long list of injured players.
“We are five points behind the leaders with 10 rounds remaining and must stay in the chasing pack for now. Who knows what might happen in the last few rounds?”
Defeat completed a miserable nine days for Pirates, who were lucky to lose only 2-0 away to trophy-holders Esperance of Tunisia in a CAF Champions League group game last week.
The beating in Tunis was the first in 28 matches in all competitions and the defeat by SuperSport the first domestic defeat in 21 games.
Jele, who was yellow-carded and misses a match against AmaZulu this Saturday, was the lone survivor from the Pirates team humiliated 6-1 by SuperSport in Nelspruit two seasons ago.
“It was not our night with the penalty miss and having three goals disallowed,” conceded the captain. “We outplayed SuperSport but they outscored us.”
All Pirates goals were cancelled for offside, but the referee should have allowed the first from Musa Nyatama as he was onside and his offside teammates were not interfering with play.
Big-screen replays confirmed that Shonga was offside before his two shots entered the net in front of a large, vibrant crowd in the 40,000-seat 2010 World Cup venue.
Recently recalled veteran Pirates goalkeeper Wayne Sandilands gave another outstanding performance with his first-half parry from a Daniels header the highlight.
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