The 2012 champion Chiefs had already clinched the New Zealand conference but needed to win the match to ensure they finished in the top two. The victory moved them to 66 points and top of the table.
The second-placed Bulls (63) play the Stormers later on Saturday, where they should know what they need to do to edge out the third placed Bumbies (59), who face the lowly Force in Perth beforehand.
The Blues were bidding farewell to All Blacks trio Ali Williams, Anthony Boric and Rene Ranger and their intensity illustrated their intention to send them off on a winning note.
That intensity boiled over, however, when loose forward Kane Barrett was sent off by referee Chris Pollock in the 23rd minute for rucking Craig Clarke too close to the Chiefs captain's head.
The Chiefs had been leading 8-3 after flyhalves Baden Kerr and Aaron Cruden traded penalties before Chiefs inside centre Bundee Aki crossed when Barrett was dismissed.
The reduction to 14 men only helped the home side, who took the lead when fullback Charles Piutau flopped on a clever Kerr grubber kick behind the defensive line.
The Chiefs then exploited their numerical advantage with a rolling maul from an attacking lineout and hooker Hika Elliott barged over.
Kerr added a penalty to lock the scores at 13-13 at half-time then he slotted his third of the game to give the home side a 16-13 lead, but he was then yellow-carded for a deliberate knockdown shortly after to give the Chiefs a two-man advantage.
The Chiefs took advantage of the sinbinning with Tameifuna grabbing his first try from in close, before he added his second shortly after Kerr returned to give the visitors their bonus point.
Replacement Gareth Anscombe added a penalty after the hooter to pad the advantage, which could have been more comfortable had Cruden not missed all four conversion attempts.
"We would like to thank the Blues for a pretty intense game. Obviously with 14 men it was going to be a difficult task for you lads and I think you showed a hell of a lot of pride that you did," Chiefs captain Clarke said in a pitchside interview.
"We've got bigger things to move on to but we are pretty chuffed (with the win)."
15 Charles Piutau, 14 Frank Halai, 13 Rene Ranger, 12 Francis Saili, 11 Lolagi Visinia, 10 Baden Kerr, 9 Piri Weepu, 8 Peter Saili/Brendon O'Connor, 7 Luke Braid, 6 Kane Barrett, 5 Ali Williams (captain), 4 Anthony Boric, 3 Angus Ta'avao, 2 Keven Mealamu, 1 Sam Prattley
Substitutes: 16 James Parsons, 17 Ofa Tu'ungafasi, 18 Liaki Moli, 19 O'Connor/Sean Polwart, 20 Jamison Gibson-Park, 21 Jackson Willison, 22 Marty McKenzie
15 Andrew Horrell, 14 Lelia Masaga, 13 Tim Nanai-Williams, 12 Bundee Aki, 11 Robbie Robinson, 10 Aaron Cruden, 9 Tawera Kerr-Barlow, 8 Matt Vant Leven, 7 Tanerau Latimer, 6 Liam Messam, 5 Brodie Retallick, 4 Craig Clarke (captain), 3 Ben Tameifuna, 2 Hika Elliot, 1 Toby Smith
Substitutes: 16 Mahonri Schwalger, 17 Pauliasi Manu, 18 Mike Fitzgerald, 19 Sam Cane, 20 August Pulu, 21 Gareth Anscombe, 22 Asaeli Tikoirotuma