Akaki Tabutsadze (26) scored a try after 90 seconds, which Tedo Abzhandadze (24) converted before adding two penalties in a dominant half-hour that suggested a 19th successive win in the fixture for the Georgians.
However, Portugal turned it around with two tries by winger Raffaele Storti (22) either side of the break and they dominated the second half.
Replacement hooker Tengizi Zamtaradze (25) levelled the score two minutes from time but Luka Matkava (22) missed the conversion and then Portugal's Nuno Sousa Guedes (28) missed a penalty shot that would have won the match.
"That’s rugby. Rugby isn’t one half," Maisashvili said. "In the first half we had a lot of opportunities to score more tries. We dominated everywhere but we didn’t execute.
"In the second half we lost everything. Portugal played very well. We almost lost this game.
"It was many reasons. The opposition took their opportunities and the reality is they caught the momentum and caught us with quick tries.
"When you are chasing a score you make a lot of mistakes. It cannot be one reason. When you have your opportunities you have to take them. We had four chances to score a try in the first half and we lost them.
"At the breakdown we lost a lot of penalties - three in the first half - and we did not fix it. In the maul we had a couple of times when we had the chance to drive but they stopped us. It was definitely the breakdown where we lost our attacking play."
Captain Merab Sharikadze (30) also bemoaned their failure to chalk up the points in the opening half-hour, when their scrum was on top and Portugal were making repeated errors.
"If you don’t capitalise on your chances, the opposition does," he said. "It’s a big lesson -- when you have a chance you need to get the maximum from it.”
After the fourth World Cup draw, and first since 2011, Georgia must now play Fiji and Wales and face a tough challenge to continue their four-tournament run of claiming at least one victory.
