In making 10 changes to his side following Friday night's 2-0 Premier League victory at home to Everton, Smith played a dangerous game, but those that came in rose to their task.
"Tonight was a big opportunity for us to give players some minutes," said Smith.
"They went out there and performed really well. It's a team that has not played together before but they understand our system, their responsibilities within the system, and they played it well.
"The finishing, the clinical nature of our game was very good, although I didn't think our general performance was all that good. I thought they gave us problems at times.
"(Crewe manager) Dave (Artell) has a really young, bright team but it was the goals that changed the game and 6-1, in the end, is a really good result for us.
"We've got a lot of minutes into a lot of lads who needed it."
Villa made the perfect start, with one of three players making a competitive debut in £12million defender Ezri Konsa heading home the opener inside four minutes.
Although Crewe had chances to equalise, two Conor Hourihane goals before half-time gave the home side no way back in the second half.
"From 2-0 I thought it would be tough for them, and we then got a third just before half-time," added Smith, who is targeting the League Cup as a silverware opportunity this season.
"We've a proud history in this competition, and we wanted to go into the game making sure our attitude was right because we knew they were a bright team.
"We showed the importance of what this competition means to us."
Striker Keinan Davis made it 4-0 in the 69th minute with his first goal for 20 months, while substitute Frederic Guilbert curled home a superb fifth soon after
Ryan Wintle deservedly netted a consolation for Crewe six minutes from time before Villa completed the rout moments later when Jack Grealish – on as a 67th-minute substitute – drove home a backheel from Henri Lansbury.
The only negative for Smith was another debutant in Matt Targett sustaining an injury that forced him off just before half-time.
Smith said: "He felt his hamstring. We got him off as quick as we could. It will probably keep him out for a week or two. I don't think it's as damaging as what it could be."
Crewe boss Artell felt his side had gifted Villa too many goals.
"We've helped them at times, looked good at times, been soft at times," said Artell.
"We thoroughly deserved a goal, so it's a bit of a mixed bag but a great barometer for our young players."