The tourists trailed 26-7 with just 12 minutes left, only for converted tries by flanker Tom Croft - his second of the game - and scrum-half Mike Phillips to set up a thrilling finale.
McGeechan said: "With eight minutes to go we were playing well. We had given ourselves a mountain to climb. I thought that in the last eight minutes we would probably have another crack and do it in the end.
"I thought we would get one more opportunity in that last five minutes and I think they did as well."
McGeechan admitted the Lions, though, gave the Springboks too great a head start.
"The penalties killed us in the end," he added. "Without playing much rugby South Africa were able to accumulate enough points."
But he backed his side to bounce back in the next two Tests.
"(I am) hugely proud of the rugby. That's high quality rugby," he said. "Of Course we can (still do this). That's why we are here."
Brian O'Driscoll said the team was disappointed their comeback had fallen just short.
He said: "With four minutes to go we still had everything to play for. We were in the ascendancy, had ball and they were panicking bit.
"So the overriding feeling is disappointment."
He added: "We scored three tries against the world champions. That will give us huge heart but essentially it's a loss."
The Irishman believes the Lions can still win the series.
"We are going to have to do it the hard way," he said. "Essentially we have got two cup finals, no margin for error. We have to have that second half performance for 80 minutes next week. We know we've got it in us."
Fellow centre Jamie Roberts felt the match was a missed opportunity for the Lions.
"That's a chance gone begging there for us," he said. "The second half was an outstanding performance to come back from that far down."
Leicester flanker Croft admitted a slow start and conceding too many penalties had cost his side dear, but nevertheless took positives from the defeat.
"We're 1-0 down but we've got two games to go and we've shown what we can do," the 23-year-old said after crossing for two tries on his Lions Test debut.
"All credit to the boys - superb performance, determination... but we've got to start well.
"The Boks are a good side. They took it to us in that first half. We gave away a couple of penalties where we shouldn't have done. Again it's that penalties thing.
"In that second half we had them up on the ropes. We kept them going forwards. We've got to take some positives out of it.
"We can take that strength into the next game. The boys are very close, we've shown what we can do. There's a massive spirit within our squad. Come the next game we'll start with a bang."
South Africa fly-half Ruan Pienaar acknowledged it had been more tense at the end than the world champions had hoped.
"It was very, very scary," he said. "We let them back at the end. The Lions played really well at the end, broke the line a few times.
"The guys showed a lot of character to keep them out just before they got over the try-line.
"It was very physical. The Lions have got great backs. Obviously we needed to work very hard in defence to keep them out."