Next man through to the last 32 was another American, Matt Kuchar. He got the better of Japan's Hiroyuki Fujita 3 and 2, while Garcia had to get up and down from just short of the 17th green to keep his nose in front.
Jaidee then fired in his approach to the last to four feet and holed it for a birdie that took the match into extra holes.
Poulter could not finish Omega Dubai Desert Classic winner Gallacher off as quickly as he hoped, making a mess of the 14th to be only two ahead, while Justin Rose, who had been four up after five on South Korean K J Choi, bogeyed the tenth on the resumption to be just one in front.
Jaidee almost eliminated Garcia on the 19th, lipping out from nine feet, while Gallacher sank a curling 15 foot birdie putt on the 15th, but it was only for a half as Poulter had pitched to three feet.
The Englishman then bunkered his tee shot to the next, but came out to a foot and remained two up, now with only two to go.
Compatriot Chris Wood resumed all square with Masters Tournament champion Bubba Watson and after bogeying the seventh was favourite to win the long next. However, the American chipped in for eagle, Wood missed and when he bogeyed again on the ninth the gap was three.
That became Lee Westwood's advantage over Spain's Rafael Cabrera Bello when he two-putted the eighth for birdie, but Rose thinned a bunker shot at the 12th and found himself all square.