Rodgers critical of missed handball
Efforts from Gary Cahill and Diego Costa overturned Emre Can's first Liverpool goal and were ultimately enough to move Jose Mourinho's side 15 points clear of the Anfield club with 11 games played.
However, Cahill - who had seen a similar infraction go unpunished in the first half - clearly handled a Steven Gerrard shot late on as Liverpool desperately sought an equaliser.
Rodgers was quick to admonish referee Anthony Taylor for what he considered a blatant error.
He told BT Sport: "I thought it was a clear penalty, we'd worked so hard in the game and obviously ended up falling behind but you need them decisions.
"It's quite obvious that the player leans into the ball and it hits his hand. It's a clear, clear penalty and the referee's got a good view of it.
"So that's very disappointing because the players gave everything in the game playing against a good side. The players deserved a point at least, that's for sure."
Though Rodgers conceded morale has been affected by Liverpool's stuttering form this season, he remains hopeful the return of Daniel Sturridge in a fortnight's time can spark life into their campaign
England striker Sturridge has been troubled by thigh and calf injuries since last representing Liverpool on August 31.
"We've got to keep working hard," Rodgers added. "I only hope we get some injured players back in that give us the threat we've had for the majority of my time here.
"We hope Daniel will be ready when he comes back after the international break.
"There's no magic formula. You've just got to work hard.
"Of course confidence can get affected when you're not winning games."