Asia failed to win a foursomes match on Sunday, but enjoyed two wins and two halves at the Empire Hotel and Country Club after trailing 3-1 at the start of play.
Europe require 3.5 more points from the eight singles matches on Sunday to claim the title for the third straight year.
YE Yang and Kim Kyung-Tae completed a 2&1 victory over Europe's Henrik Stenson and Gonzalo Fernandez-Castano.
Asia's Ryo Ishikawa and Yoshinori Fujimoto overcame the Italian brothers Edoardo and Francesco Molinari 3&1.
European skipper Jose Maria Olazabal and fellow Spaniard Miguel Angel Jimenez finished all square with Asia's Jeev Milkha Singh and Kiradech Aphibarnrat.
Nicolas Colsaerts holed a birdie on the 18th hole as he and Marcel Siem shared a point with Asia's Wu Ashun and Bae Sang-Moon.
Asia have won the Royal Trophy only once, in 2009, with Europe winning the four other editions. A fifth win looked odds on after Olazabal's side had taken three-point lead from the opening foursomes but Asia had other ideas on the second day.
India's Jeev Milkha Singh rolled in a four-foot putt on the 18th as he and Kiradech Aphibarnrat of Thailand scored a fighting half in their match against Olazabal and Spanish compatriot Miguel Angel Jimenez in the opening game of the day.
Singh's putt also drew a welcome response from the crowd on his 41st birthday.
"It was nice to hear the crowd singing happy birthday. They might not have bothered if I had missed on 18," the Indian said in a television interview.
Jimenez and Olazabal, who was nursing what appeared to be a neck injury throughout the round, both narrowly missed makeable putts on the last to claim the win.
Despite the pain, Olazabal, who grimaced after hitting shots and had his caddie help stretch out his shoulders and neck during the round, will lead off the singles against South Korean Bae Sang-moon on Sunday.
South Korean Kim Kyung-tae then rolled in a four-footer on the 17th as he and compatriot Yang Yong-eun claimed a topsy-turvy 2&1 win over Spain's Gonzalo Fernandez-Castano and Swede Henrik Stenson.
The Korean duo were five up through nine holes but the Europeans closed the deficit and trailed by only one with three to play before Kim's pinpoint approach to the par-four 17th set up a birdie and Asia's first win of the sixth edition.
Bae could have added the second win for Asia shortly after but he missed his birdie effort on the 18th to half the match with Belgian Nicolas Colsaerts and German playing partner Marcel Siem.
Bae and Chinese partner Wu Ashun would have had two putts for the win but European Ryder Cup winner Colsaerts rolled in a 14ft effort on the final green to set up the half.
Japanese duo Ryo Ishikawa and Yoshinori Fujimoto then sealed a come-from-behind 3&1 win over Italian brothers Edoardo and Francesco Molinari to make it Asia's day and give them hope of winning the trophy.
"The whole team knows we have to play our very best golf to win, but we have given ourselves a good chance," Wu said.
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- Ryo Ishikawa
- Jeev Milkha Singh