Steenson kicked six penalties and two drop goals for a 24-point haul
Exeter stand-off Gareth Steenson booted the Devon side into the Premiership with a flawless kicking display to stun Bristol at the Memorial Stadium.
Starting 9-6 behind in the Championship play-off final second leg, Bristol were rocked by two early Steenson penalties.
The home side then got to within two points on the night on 16 minutes thanks to Luke Arscott's converted try.
But Steenson took over, kicking six penalties and two drop goals before sub Simon Alcott's last-gasp clinching try.
This is the first time that a place in the Premiership has been decided by a play-off since 2000. And the RFU Management Board had already revealed earlier in the day that both Bristol, relegated from the top flight a year ago, and Exeter had already met the authorities' Minimum Standards criteria.
So there will unquestionably be Premiership rugby at Sandy Park for the first time come September.
Steenson ended with a 24-point haul, keeping his nerve to land perhaps the most crucial of his kicks, just seconds after opposite number Adrian Jarvis's effort had hit the post at the other end, midway through the second half when there were still just eight points in it.
The Irishman had also been in form with the boot on Exeter's last trip to the Memorial Stadium back in October, kicking 15 points in a 35-26 win.
And it was his three penalties at Sandy Park last week which edged Rob Baxter's side into their narrow first-leg 9-6 lead.
Steenson delighted after play-off triumph
But, while Steenson's boot this time mercilessly punished a poor, indisciplined display by Bristol, who made a lot of handling areas and struggled at the line-out, the bottom line was that the Chiefs just seemed to want it more.
Exeter started at a frantic pace, not allowing their hosts to settle, and were rewarded by two Steenson penalties inside 13 minutes.
With Exeter now nine points ahead on aggregate, Bristol sprung into life, and several phases of attacking play ended with Jarvis slipping through the Chiefs defence to release Arscott for the game's first try.
Brother Tom Arscott put the conversion over to earn a 7-6 lead on the night and send the noise level soaring inside the Memorial Stadium as the home fans in a Championship record 11,850 crowd sensed their side were beginning to gain a foothold.
But Bristol's optimism was to be short-lived as indiscipline handed Exeter two more penalties in quick succession, with the ever reliable Steenson earning a 12-7 lead.
Hull gutted to miss out on promotion
Rain began to fall heavily, and handling errors started to creep in but Steenson's kicking remained flawless. And, when Exeter made it back into their opponents 22 on the half-hour mark, the Irishman kicked a vital drop goal for a 15-7 interval lead.
There was no let up in intensity after the restart, but Bristol were equal to everything their visitors threw at them, scoring the first points of the second half when Jarvis slotted a 54th minute penalty for 15-10.
Jarvis then went so close to reducing the overall arrears to five points when his next penalty attempt agonisingly bounced back off the upright. And the Chiefs instantly took full advantage with another penalty of their own, Steenson coolly restoring the 11-point aggregate lead.
Paul Hull's side threw everything at Exeter but, as had happened all night, the ball kept spilling loose at key moments.
I'm very proud of the players - Baxter
With Bristol camped inside their opponents 22, Steenson was again the Chiefs hero, this time making a superb break out of defence. And, when play broke down, the fly-half stuck yet another penalty between the sticks to make it 21-10 on the night.
Steenson appeared to have rounded off a superb performance with a second drop goal in the 78th minute which sent the visiting support into raptures as they began to realise the Premiership dream was just seconds away.
But there was still time for Exeter to cross the Bristol line for the first time in the tie, Alcott being adjudged by the video referee to have gone over from the final play.
Steenson's conversion went wide but the talismanic fly-half had already done enough to ensure top-flight rugby will be enjoyed by rugby fans in Devon for the first-time.
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