The Swans came from two goals down to equalise in the first half as efforts from Jonjo Shelvey and Wilfried Bony cancelled out Daniel Sturridge and Jordan Henderson's early strikes.
Sturridge restored Liverpool's lead after 36 minutes but they were pegged back once again as Bony dispatched from the spot just after the break before Henderson netted a 74th-minute winner.
Monk lamented the Swans' laboured start to the game, but says that he was left with plenty of positives following an impressive second-half showing.
"We didn't start well. We didn't get the ball moving, we were too slow, we had too many touches," he told Sky Sports.
"I told the boys that at half-time. It's just that belief.
"Coming to a team that are trying to compete in winning the league and are right up there with the quality that they've got, we never really got going.
"But I think in the second half, we had a right go and I thought we deserved something from the game."
Due to cup commitments, the tie was Swansea's sixth game in the space of three weeks and Monk was pleased that it did not show in the performance.
"With the amount of games we've had recently, the fact that they kept up the energy levels right to the end was magnificent," he said.
"I think if we'd started like that, the way we did in the second half, it would have been a different game.
"That's the character we've got. I don't ever worry about that. It's always been there in the group, it's just bringing it out of them and I think it's coming and it's getting better for the boys.
"To come to a place like this, away from home, where they're destroying teams week-in-week-out, it's all credit to my players, they totally deserved something from the game.
Speaking of Henderson's winner, Monk said: "The fourth goal was a ricochet back into their path and sometimes you can't do anything about it. You create your own luck.
"In the second half, I knew that if we did it properly, we could get back into the game and we did that. We got the goal and we were looking a threat.
"Towards the end, you ask them to keep on pushing, I think that's the point you need to be either winning the game or drawing the game.
"We were a credit, we went to the very end again. I can't be prouder of the boys. It was a monumental effort. With the amount of games we've had, it's all credit to them."