The BVB trainer was disappointed that the referee chose not to dismiss the midfielder for a second bookable offence at Signal Iduna Park
BVB, who finished runners up last season, won 2-0 on Tuesday thanks to a first-half double from Marco Reus but a 3-0 loss at the Santiago Bernabeu condemned them to a quarter-final elimination at the hands of Carlo Ancelotti's side.
Alonso's persistent fouling in the opening 45 earned him a deserved booking, but Signal Iduna Park supporters were incensed when the Spaniard was not given his marching orders for a foul in the second half on Henrikh Mkhitaryan and Klopp felt their indignation was totally justified.
"I appreciate tremendously Xabi Alonso is a one of the best players in the world," the BVB boss told reporters, "but what he did was a tactical foul that involves yellow card and already had one. It was red-card offence.
"That would probably have complicated things by Real Madrid. But whatever I say will not change anything now.
"We were the better team but only won 2-0 and we're out - that's life."
Real Madrid have reached the semi-final stage for the fourth campaign in a row, although on all three prior occasions they have failed to reach the tournament decider.