Buttler, who hit an unbeaten 106 in Rajasthan's win in the second qualifier, finished this season as the leading batsman with 863 runs and his wicket brought the house down at the nearly full 132,000 capacity stadium.
Afghanistan's Rashid Khan returned impressive figures of 1-18 from his four overs of leg-spin to keep down the opposition scoring.
Pandya, who sent back his opposite number Sanju Samson for 14, kept up the pressure and got the big-hitting Shimron Hetmyer out for 11 with a caught and bowled.
Fast bowlers Mohammed Shami and Yash Dayal, who got opener Yashasvi Jaiswal for 22, worked up good pace to keep Rajasthan, who elected to bat first, on the backfoot from the start.
Gujarat need a modest 131 to win the IPL in their debut season after they became one of the two new teams alongside Lucknow Super Giants added to the world's richest cricket tournament.