After being reduced to 51/4, Karthik (52; 38b; 4x4 2x6) and Shubman Gill (28; 17b; 3x4 1x6) got together for a 55-run fifth wicket stand before Andre Russell smoked an unbeaten 25-ball 49 to help the purple brigade post 169/7 in 20 overs.
Chasing 170, Royals were 47/0 inside five overs and then Sanju Samson (50; 38b; 4x4, 2x6) and Ajinkya Rahane (46; 41b; 4x4, 1x6) set the platform for a victory before the pair were removed by Piyush Chawla (2/24) and Kuldeep Yadav (1/18) respectively.
Royals could never recover from there as their thin middle order got exposed badly. They could manage only 144/4 in 20 overs.
"I think winning four games on the row feels good. Especially this one as we were under the pump," Karthik said at the post-match presentation ceremony. KKR came into the game on the back of a three-game winning run.
"Credit to Shubman Gill. He absorbed all the pressure when he walked in. Before he got out, he played some lovely shots and eased the pressure off me.
"In these games, it is about the attitude and the belief about how much you have got. In this kind of crunch games, it is never about the par score; it's about how much pressure you can put on the other side," he added.
Praising their bowlers, Karthik said Royals always think "out of the box" and gets the opposing captain thinking.
"The bowlers have picked up their game. We fielded well as well today, which was good to see. They (RR) always think out of the box. They kept a short leg and put pressure on the batsman. They get the captain thinking. You need to hold your game and nerves. Every game is important from here on. Next game is important as there are two good teams which will be competing."
Meanwhile, a dejected Rajasthan Royals (RR) skipper Ajinkya Rahane blamed his team's inability to finish off the game after getting a good start.
"Disappointed, especially given how we started, taking 4-5 wickets early. That partnership (between Karthik and Gill) and the dropped catch of Russell hurt us. When you get a start like that, you generally look to win the game," he said.
"I think KKR bowled really well. They know the conditions really well. It was gettable. They were 10-12 runs short. It's important to finish the games and we didn't have partnerships towards the end.
"When I and Sanju were batting, we were looking to play positively. I told him during the second timeout that I will go after bowlers and that you bat through. But unfortunately, he got out. I think overall we played good cricket. Our bowling unit fantastic this season. Need to improve our batting and fielding," he added.
KKR will now meet Sunrisers Hyderabad (SRH) here on Friday to decide who faces Chennai Super Kings (CSK) in the final on Sunday.