Djuricic's second penalty appeal came when he appeared to be fouled by Joe Allen, but again referee Kevin Friend ignored his appeals.
Then Can handled in the penalty area as he tried to control before clearing a cross, yet once more Friend waved play on.
Liverpool took advantage of more good fortune to seal the points in the 73rd minute when Southampton's Matt Targett slipped while trying to clear and the ball fell perfectly for Sterling to slot a deflected shot past Fraser Forster.
"I call it unlucky in the decisions of the referee. The second situation is a clear penalty to Filip Djuricic," Saints boss Ronald Koeman said.
"We have more ball domination but we create four or five good positions to score and referee didn't whistle for a clear penalty.
"We played much better than Liverpool. Sometimes you need that luck on your side."
Rodgers added: "The player has gone over far too easy, no way that was a penalty.
"I thought Kevin Friend was excellent and I think all his decisions were right.
"We knew it would be a huge result to win here and it was."
Elsewhere yesterday, Mauricio Pochettino praised the fighting spirit of his Tottenham Hotspur players after Harry Kane rescued a dramatic 2-2 draw against West Ham United.
Pochettino's team had fallen 0-2 behind at White Hart Lane as the Hammers struck through Cheikhou Kouyate and Diafra Sakho, but an error by goalkeeper Adrian allowed Danny Rose to get one back with nine minutes left.
And when in-form striker Kane was fouled in the area by Alex Song deep into second-half stoppage-time, he kept his composure to smash in the rebound after his spot-kick was saved for his 24th goal of a breakthrough campaign.
"We showed character, that's the most important thing," Pochettino said. "Going into the last 10 minutes at 2-0 down, to finish the game 2-2 is something to be proud of."
However, West Ham manager Sam Allardyce, frustrated that Kane went down easily under Song's challenge for the penalty, claimed Tottenham were lucky to take a point from a pulsating London derby.
"They're so lucky today," he fumed after the match.
"You can argue whether Alex Song has made enough contact for Harry Kane to go down, but when you're desperate like Tottenham are, you get touched and the player is going to fall over," Allardyce said. AFP