The Londoners came close to a second-round exit as Birkirkara won 1-0 on the night to force extra-time and then penalties, but all five visiting takers scored to secure a 5-3 victory.
James Tomkins was sent off just before half-time, and Bilic was impressed with a backs-to-the-wall performance that set up a tie with Astra Giurgiu of Romania who saw over Inverness Caledonian Thistle.
Mark Noble, Mauro Zarate, Aaron Cresswell, Joey O’Brien and Diego Poyet were West Ham’s successful penalty-takers, and Bilic said: “They were very confident when I asked them.
“That is good because I have seen the character and confidence in their eyes. They were all brilliantly taken. We now go to Romania and it will be the same kind of atmosphere.
“The most important thing is that we are through. They started better and they were very aggressive. But after they scored we took control of the game.
“We had a man sent off and we had to adjust to that. But the guys showed their spirit and commitment and the most important thing is that we got no injuries and we have got through.”