The Chiefs still advance to the last four under the "lucky losers" rule, but it was do-or-die on Saturday for the sixth-placed Blues who produced a suitably desperate performance.
A try from centre Daniel Rona and 14 points from the boot of Damian McKenzie looked to have earned the Chiefs the win but the Blues pounded away at the line in a frantic finish before lock Beehre reached over to score and Beauden Barrett converted.
"There's been a whole lot of ups and downs through our season, and... although there's another week for us, we've got to enjoy that kind of moment," said Blues skipper Patrick Tuipulotu.
"I'm glad we were able to sort of grind it out in that last 10 minutes."
The match at Waikato Stadium was dominated by a titanic forward battle complemented by a dazzling duel between All Blacks playmakers McKenzie and Barrett.
Three penalties from McKenzie to one from Barrett gave the Chiefs a 9-3 lead at halftime after both sides bombed what looked like certain tries.
Chiefs centre Rona burst down the left wing in the 27th minute and while his kick infield looked to have given Cortez Ratima a try on a plate, the scrumhalf knocked on with the line at his mercy.
McKenzie gave the Chiefs a 6-3 lead with his second penalty four minutes later and hit the turf soon afterwards after being tripped by Ricky Riccitelli while chasing a Shaun Stevenson chip through.
That earned the hooker 10 minutes in the sin bin and McKenzie added his third three-pointer from the resultant penalty but Barrett was not done with his contribution to the first half.
The twice World Player of the year chipped the ball over the home defensive line at midfield, got to it first and volleyed it past the last defenders but was unable to get his hands on it under the posts.
Barrett cut the deficit with his second penalty soon after the break but McKenzie responded in kind three minutes later and the Chiefs were 19-6 ahead just before the hour mark.
The home side laid siege to the Blues try line and a superbly judged pass from McKenzie sent Rona through a gap in the defensive line for the first try of the game.
Caleb Clarke got across the try line for the Blues minutes later only to have the ball knocked out of his grasp but forward Kurt Eklund was able to force his way over for a converted try that cut the deficit to 19-13 with 15 minutes to play.
Blues number eight Hoskins Sotutu was denied what would have been a winning try when he was called back for crawling and it was left to Beehre to secure the victory.
"I thought we had that game in the bag," said Chiefs lock Tupou Vaa'i. "I guess footy's a strange game sometimes and you can start on a high and then it can humble you real quick.
"I guess we've still got a life now, and our season's not over yet."
