Hammers captain Jarrod Bowen swapped goals with Brentford striker Igor Thiago as the two teams finished extra-time at 2-2.
In the shootout, Ouattara fluffed his Panenka attempt for Brentford, so seeing West Ham win on penalties, 5-3.
Andrews said afterwards: "It was a proper cup tie. They took the lead and in the first half we didn't hit the heights we could and should've done but in the second half we got back into the game.
"Extra time, we huffed and puffed and when you get to penalty kicks, it is not a toss of a coin but you need luck. They took theirs very well and unfortunately we missed one.
"With a goalkeeper like Caoimhin, you feel he will save one and their penalties have to be good and they were."
On Ouattara's missed Panenka, Andrews reasoned: "When you miss a penalty it is a tough moment, but it is easier to not take a penalty than take a penalty. We've had it in the past when people have been persecuted and ridiculed for missing a penalty on the biggest stage and that's ridiculous.
"He practices that technique a lot and will have the utmost support from me and everyone around.
"We really wanted to get through to the quarter-final but it was not to be. But we have nine games left where we can push on and do something special."