Exeter reached the Heineken Champions Cup quarter-finals on try count after extra-time following a dramatic 33-33 draw with Montpellier at Sandy Park.
Hooker Jack Yeandle's touchdown in the final seconds of 20 minutes' added time put Exeter within striking distance, then Joe Simmonds' nerveless conversion meant Exeter progressed 5-4 on tries.
French champions Montpellier, despite having their England international number eight Zach Mercer sent off following a high challenge 10 minutes after half-time, dug deep to almost turn a pulsating clash their way.
Fly-half Paolo Garbisi levelled the game at 26-26 through a long-range penalty with the final kick of normal time, then Montpellier replacement Louis Carbonel touched down midway through the second period in extra-time, and Garbisi converted, but Yeandle and Simmonds had the final say.
Exeter looked well set after tries from Scott Sio, Sam Simmonds, who will join Montpellier next season, Tom Wyatt and Josh Iosefa-Scott left the visitors reeling, while Joe Simmonds kicked four conversions.
Exeter impressively wiped out an early 12-point deficit given to them through tries by replacement hooker Curtis Langdon and wing Vincent Rattez, while Garbisi ended with three penalties and two conversions, with lock Lenni Nouchi also touching down.
It was an outstanding effort by Exeter, especially given their early difficulties in counteracting Montpellier's physical approach.
The Chiefs will make a fourth Champions Cup quarter-final appearance - they host South African challengers the Stormers next weekend - as they continue to put erratic domestic form to one side in pursuit of a second European title.
Saracens pounced on a terrible blunder by the Ospreys to clinch a 35-20 victory at StoneX Stadium that secured their place in the quarter-finals.
With a desperately-tight round-of-16 tie locked at 20-20 approaching the final 10 minutes, Rhys Webb took a quick line-out around his 22 only for the ball to fall straight into the arms of Duncan Taylor.
It was Taylor's first touch of the afternoon having just arrived as replacement for Alex Lozowksi and the former Scotland centre strolled over before being mobbed by his team-mates.
League form pointed to a comfortable home win with Saracens leading the Premiership and Ospreys sat 12th in the United Rugby Championship, but the only Welsh side left in the knockout phase instead went toe-to-toe in a pulsating clash.
They lost wing George North shortly before kick-off but still paraded nine internationals and it was not until Max Malins claimed the second of his two tries in the 57th minute that Saracens were back in the hunt.
It was a far-from-vintage display from the three-time European champions but they had the resilience to set up a last appointment with Ronan O'Gara's La Rochelle next weekend, becoming the third English club behind Leicester - who will face Leinster on Friday - and Exeter to reach that stage.
Tom Woolstencroft's late try distorted the final scoreline, which from an early stage, seemed beyond Mark McCall's men.
Toulouse turned on the style in the second half to beat Bulls 33-9 and reach the next round.
Tries from Emmanuel Meafou, Matthis Lebel and Thibaud Flament, all after the break and all converted by Thomas Ramos, took the game away from the South Africans after a cagey first half.
The hosts led 12-6 at the break, with all the points coming from the boots of Ramos and Chris Smith in a battle of penalties, but it was a different story in the second half as Toulouse's dominance showed.
Meafou got the first in the 51st minute as Toulouse cashed in on a penalty with some quick movement, and Bulls had little opportunity to recover before Lebel got the second, set up by Antoine Dupont off a scrum.
Dupont was involved again for the third, kicking the ball over for Flament to finish.
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