The defending champions are now level on points at the top of Group E with Shakhtar, and Cahill believes that their position is partly down to their never-say-die attitude.
"Winning [the match] eases the pressure on us a little bit," said Cahill. "That's what we needed to do, especially with [Shakhtar Donetsk] sitting on top of the table. It's important we got those points back.
"Thankfully Victor [Moses] got a massively important goal for us at the end. It was the last kick of the game, so it was amazing really.
"We, sort of, don't know when we're beaten at times this season. We just keep plugging away and with the creative players that we've got we're always going to create chances, so it was pleasing. The lads are buzzing. They were a tough, tough team to face. In the first game they were fantastic. I think they shocked a lot of people with how much quality they've got, and tonight was no different."
Chelsea host Liverpool in their next match on Sunday.