The tourists won 22-17 in an attritional contest in Cape Town, ending the game strongly after trailing 12-3 at the interval.
“The message at half-time was that we are still in the arm-wrestle,” said Gatland, the Lions head coach.
“If Dan Biggar had kicked that penalty just before the break it would have been 12-6. Instead, it was 12-3 but we knew we had given away quite a few soft penalties to allow them opportunities.
“So, we knew we just had to keep our patience because we will get chances so don’t try to force things – and as the second half went on I thought we got stronger and stronger to get back into the game.
“It was a really tough, tight Test match and with the bounce of the ball it could have gone either way, but thankfully we came from behind and finished really strongly.”
Luke Cowan-Dickie scored the breakthrough try for the Lions but South Africa responded quickly with one of their own from Faf de Klerk. Biggar kicked 14 points for the visitors and replacement Owen Farrell added a late penalty.
Gatland expects the Springboks to be fired up for the second Test, also in Cape Town.
“They will be hurting from this because they are an incredibly proud nation and world champions, so next week is going to be even bigger and tougher, I would expect.
“But from our point of view, you win that first one and you know that whatever happens you are going to that last game of the series, so that keeps everyone engaged and really interested in it.
“The message to the players is that this was not just about the Test match 23. The non-playing guys did a brilliant job preparing the Test guys this week for the game, so the victory was as much about them and the whole squad as the guys who took the field.”
South Africa had no complaints about the officials’ decision to chalk off Willie le Roux’s potentially match-changing try.
The Springboks saw possible scores for both le Roux and Damian de Allende ruled out, but head coach Jacques Nienaber refused to offer any criticism of the officials.
The Lions had been angry with the appointment of South African Marius Jonker as television match official (TMO) ahead of the contest, but it was his intervention that judged le Roux offside, denying the Boks full-back the try.
“I thought it was tight. As soon as we saw the try was given [by the match referee] we, as coaches, thought it was going to be extremely tight,” said Nienaber of the Le Roux decision.
“But I completely agree with and trust the decision they made. That is their profession, that is what they are good at.
“It could have gone both ways in my opinion, but I 100 per cent agree with the TMO decision.
“Sometimes those inches go for you and you score a brilliant try from a counter attack and sometimes it goes against you.”