Gil, making his first start since arriving from Valencia earlier this month, opened the scoring with a stunning 51st-minute strike on Sunday, before Andreas Weimann added a second 20 minutes later.
Substitute Callum Wilson netted a late consolation for the Championship leaders, but Lambert was relieved to see his side into the fifth round.
"We needed to try to get through," he was quoted as saying by BBC Sport. "All credit to the team. I thought we deserved the win.
"Gil is lighting the place up and it was a world-class goal. Being a foreign player you can take time to adapt or hit the ground running and he has done the latter.
"He does the same in training. In training you can't get the ball from him as he has a low centre of gravity.
"I'm delighted to get through. It's a big win."
Villa had scored just one goal in their previous six matches in all competitions prior to Sunday's win, and Lambert says he can see the light at the end of the tunnel after that slump.
"The supporters went through a hard time just like the players," he added. "The win today will certainly help us.
"We took a lot of knocks but all credit to the lads, they never let their heads down, even in training."
