The Welshman held on for a 21-18 victory after 25 ends to move into the semi-finals.
In the women's event, third seed Vanessa Atkinson moved comfortably into the semi-finals for the second time.
But the late sin-binning of Bath replacement Stuart Hooper allowed Blair to strike his fourth penalty and usher the Blues into the semi-finals of the competition.
With the match on a knife-edge, Evans had a late attempt at a drop-goal to put Quins into the semi-finals, but his effort was wide of the posts.
With the match on a knife-edge, Evans had a late attempt at a drop-goal to put Quins into the semi-finals - but his effort was wide of the posts.
England must now beat India on Sunday, and West Indies the following day, but even two positive results would not necessarily be enough to put the hosts into the semi-finals.
But the fly-half, who missed five kicks at goal in the Basque country, saw his effort fall short as Biarritz surged into the semi-finals for the first time since 2006.
Carmarthen almost found an immediate reply as Cardiff went to sleep at the back, but Forde was on hand again to frustrate the home side and secure the Bluebirds' passage into the semi-finals.
BBC: SPORT | Football | Welsh | Carmarthen 2-3 Cardiff (aet)
But just two months later, following a 3-3 draw away to Red Star Belgrade which took his side into the semi-finals of the European Cup, Morgans was aboard the AS-57 Ambassador which crashed on take-off from Munich airport on the 6 February 1958.
Only two are allowed to progress into the semi-finals, Marleen Veldhuis, who anchored the relay team to victory on Tuesday showed every sign of breaking the world record in the semi-finals but went out more slowly this time and was well shy of Britta Steffen's mark.
But Young refused to get carried away and said now all the region's attentions would be concentrated on getting a result against Bath next week and making it into the EDF semi-finals.
Making the final cut was a shock Denise Van Outen, who had been voted into the dance-off during the quarter and semi-finals.
Blues coach Dai Young has hailed Williams' influence in the region's charge into the EDF Energy Cup semi-finals and European quarter-finals.
The top four clubs at the end of the regular campaign are expected to enter into semi-finals and a final to determine the overall champion.
The Welsh region lost 15-24 at home to Overmach Parma to drop into the second tier Challenge Cup, but did make the semi-finals of the competition before losing to Clermont Auvergne.
The eight teams taking part have been divided into two groups, with the top two from each reaching the semi-finals.
Fans ignored the danger of car bombs (which had killed 50 after the Iraqi victory in the semi-finals) as well as the risk of falling bullets (fired into the air to celebrate despite a ban) and poured onto the streets.
Kucera, who had lost to Canas in the semi-finals last year, breezed into the last four after a 6-1 7-5 win.
World number two Rafael Nadal , who Murray is seeded to meet in the semi-finals, was given a walkover into the next round when his opponent Olivier Rochus pulled out with a foot injury.
The world's top eight players have been divided into two groups for this tournament, with the top two from each progressing to the semi-finals, although world number three Serena Williams and number five Venus Williams are absent this year through injury.
But she says she will go into Friday's semi-finals (2000 BST) in better shape than before the Worlds last summer.
The teams will be divided into two groups of four teams each, with the top two teams from each group qualifying for the semi-finals.
The Intercontinental Cup involves 12 teams, divided into four groups of three, playing a series of three-day matches with the winners progressing to the semi-finals in the United Arab Emirates in November.
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