Head coach Gatland saw his team beaten 19-16 by a giant South African side as they just fell short of setting up a meeting with England in the last two.
Three penalties from Dan Biggar and a Josh Adams try which was converted by Leigh Halfpenny kept Wales in the contest, but ultimately it was not enough.
“It was a tough encounter. We knew it would be an arm wrestle. We stayed in that arm wrestle and got back into it,” said Gatland.
“There was a bit of a momentum shift in that last five minutes and it was hard to get going. Unfortunately we’ve been penalised and the game has got away from us.
“I’m not taking anything away from South Africa, I thought they played really well. They were strong and physical, their scrum was good, and they drove pretty well.
“We knew they would take us on up front. There wasn’t a lot of flowing rugby played. We tried to move the ball a little bit, particularly in that first half, but when you’re in the arm wrestle it’s about attrition. I’m very proud of the boys for not giving up and staying in there.
“With a little bit of luck it might have been different.
“I’m very disappointed, but incredibly proud of what this group of players has done at this World Cup. We can hold our heads high and leave Japan with a lot of respect.”
Gatland continued: “The support we’ve had from Wales has been outstanding. We’ve loved our time in Japan and the hospitality has been incredible. It’s important that you’re humble.
“I thought South Africa did well tonight and they deserved to win the game. As a coach you have to take that on the chin. You’re disappointed, but it’s important that you recognise when another team has played well in a close game.”
Wales were 9-6 down at the break as Biggar kicked two penalties to one from opposite number Handre Pollard. In the second half Damian de Allende scored a crucial Springbok try.
Pollard converted the effort, but still Wales refused to go away as they showed tremendous character. They opted for a scrum instead of kicking a penalty and hands from Tomos Williams and Jonathan Davies sent Adams over in the corner.
It was the winger’s sixth try of the tournament which makes him the leading marksman in Japan with two games left in the tournament. Adams has now equalled the Welsh World Cup try record of six – a feat Shane Williams managed at the 2007 tournament in France.
Wales are hurting and Pollard’s late penalty sent them to defeat, but their attentions now shift to a short turnaround for Friday’s bronze medal match with New Zealand.
The All Blacks are the only side Kiwi-born Gatland hasn’t beaten in his 12-year spell with Wales and the game will be his last in charge after a remarkable spell at the helm.
“We’ve got another game to go, we play the All Blacks. My first game in charge was against England and the dream was for it to be my last game, but it’s not to be,” Gatland said.
“This is a great group of players that we’ve been involved with. We need to recover well over the next couple of days and start preparing for the All Blacks.
“We’re really disappointed about not being in the final, but we’ve got to be proud of ourselves.
“Friday will be our last time together so we’re going to make sure we enjoy this week.
“The All Blacks are probably hurting as much as we are at the moment. The third and fourth play-off is a tough game to play, but you have to give it the respect it deserves.
“We have to go out there and do the best we can to try and win it. It’s my last game in charge against the All Blacks. It will be monumental. As a coach they are the only team I haven’t beaten with Wales. It would be nice to be able to achieve that.”
