Five-time champions Chennai restricted Rajasthan to 141-5, a total they overhauled with five wickets and 10 balls to spare in their last home game at MA Chidambaram Stadium.
Chennai moved to 14 points and third place with seven wins in 13 matches. Sunrisers Hyderabad, also on 14, slipped to fourth, while Delhi Capitals and Lucknow Super Giants remain close with 12 points each.
Rajasthan remain second in the 10-team table led by Kolkata Knight Riders, who are the only team to have already qualified for the playoffs.
The top four teams will make the playoffs with the final on May 26th in Chennai.
"Very normal to keep thinking about the qualification at this stage. The conversations we are having is to control what we can do," Rajasthan skipper Sanju Samson said.
"At the team brief, I will probably ask them to continue the process and hoping to get on the better side next game."
Gaikwad anchored Chennai's tricky chase as they lost wickets at regular intervals including when Ravindra Jadeja was called out for "obstructing the field."
Jadeja was sent back by Gaikwad while attempting a second run and he changed direction on the pitch when the throw from wicketkeeper Samson hit his hand and the third umpire ruled him out after an appeal from the opposition.
Gaikwad kept his cool and put on an unbeaten 24-run stand with impact substitute Sameer Rizvi, who hit the winning boundary.
Gaikwad is second in the season batting chart behind Royal Challengers Bengaluru's Virat Kohli (634) with 583 runs including one century and four half-tons.
Gaikwad's fellow opener Rachin Ravindra, who hit 27, and another New Zealand batsman Daryl Mitchell, who made 22, also contributed.
Earlier disciplined bowling kept Rajasthan down to a below-par total on a slow pitch as Simarjeet returned figures of 3-26 and fellow medium-pace bowler Tushar Deshpande took two wickets.
Riyan Parag top-scored for Rajasthan with an unbeaten 47.
Yashasvi Jaiswal, who made 24, and Jos Buttler, who made 21, started cautiously but the innings suffered from a lack of partnerships.