The skipper also hailed manager Jürgen Klopp and said that his managerial leadership has helped to improve the never-say-day attitude being harnessed in the squad.
Palace midfielder Joe Ledley opened the scoring for the home side just minutes into the second-half, and things continued to decline for the Reds as vice captain James Milner was shown a second yellow with 30 minutes remaining.
But Klopp's side picked themselves up, dusted themselves off and took the fight to Alan Pardew's men as Roberto Firmino pounced on a goalkeeping error from Alex McCarthy to level the scoring with 18 minutes to play.
Late on in stoppage time, substitute Christian Benteke was handed a controversial penalty as Damien Delaney went to ground and despite pulling out, his trailing knee caught the foot of the Belgian, who then proceeded to calmly roll the ball home from the 12-yard spot.
Henderson on the victory
Liverpool fought hard throughout the match, but only seemed to get going once they came under fire after going 1-0 down and former Sunderland man Henderson shared this view as he gave his thoughts post match.
The captain said that the Premier League contest was "tough, especially in the first half" as his side came under a barrage of attack from the home side, which he described as a "battle."
He said that upon scoring the first goal, his teammates "grew in confidence" which he believed meant his side "deserved all three points" thanks to the "chances we created."
Klopp said in a previous press conference that his players tended to give up towards the end, but according to Henderson that is now in the past, as he said that Klopp's ideology has been "drummed" into himself and the players to "never give up" and "always go right till the end."
On the penalty
Forward Benteke tumbled over with just seconds to go until the final whistle, as a clumsy challenge from Palace defender Delaney brough him down.
Despite protests from the home players, a penalty was awarded by the assistant official after Andre Marriner had been hesitant to award the spot-kick - which Liverpool's No.9 then converted to hand his side all three points.
Henderson said Benteke was "full of confidence" as he saw the striker step up to the 12-yard spot, as he has "seen him take penalties before in a shootout" and also in "training," he said the Belgian international was "very cool."
On Manchester City victory
The Reds went into the Palace clash having managed to score a modicum of revenge over the blue half of Manchester at Anfield on Wednesday, following on from Sunday's defeat at the hands of Manuel Pelligrini's men in the final of the Capital One cup.
The 25-year-old said the 3-0 win over Manchester City, had "gave us a lot of confidence" which he felt enabled himself and his teammates to believe that they could "win the game."
The England international said the win granted his side the ability to "create some good chances" from the victory that he described as "a good win."