Karim Benzema scored twice after Cristiano Ronaldo's opener to consign the Reds to a second successive defeat in Europe's premier competition, and the Northern Irishman admitted that Carlo Ancelotti's team were the better side.
"The first 20-22 minutes of the game, I thought we made an excellent start," Rodgers told Sky Sports News. "And then from that point through to half time [Real Madrid] showed their real quality – they were terrific.
"Second half, we're then obviously playing for pride and I thought [my players] were outstanding in the second half, they kept going and they're playing against a top-quality team and we were probably unfortunate to be 3-0 down at the break.
"I can't ask for more, that opening period we were very good and probably needed a goal in that period. You can see why they're [European] champions. Set-pieces have been an issue for us this season and we didn't defend well enough."
Los Blancos host Rodgers's squad in the next round of fixtures.