Juve played most of the second half with 10 men following the dismissal of centre back Giorgio Chiellini and although they rarely looked like forcing an equaliser they avoided capitulating tamely with a battling display.
"Unfortunately tonight we were beaten in a match that was conditioned by one or two incidents," Conte said.
"But what I am more interested in is that the lads showed that we can compete on a stage like this against a Real side that is fighting to win the Champions League," he added.
"Now it's going to be an uphill battle but playing like we did tonight we are capable of coming back and qualifying."
Real top Group B with a perfect nine points from three of six games, with Galatasaray on four, Juve on two and FC Copenhagen bottom on one.
Juve's next match is at home to Real on November 5, when Copenhagen host Galatasaray.
"When it was eleven against eleven we played well," Juve's France midfielder Paul Pogba said.
"We could have equalised in the second half but everything became more difficult after the red card," he added.
"We knew we had to defend well and restart quickly when we had the chance and we did it very well in the first half.
"We are disappointed with the defeat but it's not over and we will try to win the next game.