Boro had to come from two goals down to level things in added time at the end of the 90 minutes but League Two Crewe still managed to progress courtesy of a 4-2 penalty shoot-out win on Teesside.
Artell had seen his players hit the break with the lead after Chris Porter's 41st minute goal was followed by Charlie Kirk's finish four minutes later.
Middlesbrough, who struggled for an hour, then found their feet more to level when substitute Ashley Fletcher's 75th minute strike was followed by Marc Bola's first for the club in the first minute of stoppage-time.
But Crewe, who scored all their spot-kicks, progressed after Britt Assombalonga and Marcus Browne's poor penalties for Boro left boss Jonathan Woodgate frustrated.
Artell said: "I think we deserved to win in 90 minutes. This was a three-quarter empty stadium and it's about mental toughness as much as ability.
"We all want to score, we all want to win, we have technical players, they are not scarred because they are young, it's about believing you are going to score and this was a terrific night's football from the boys.
"I have just been told about Villa, I am delighted. It was great reward for a big night's work.
"They brought the big guns on if you like, even though they already had the big guns on, but it showed how good we were. We deserved to win.
"I looked at the team-sheet, we knew there would be changes to Saturday from them. I wasn't sure how many.
"(Anfernee) Dijksteel, (Marc) Bola, are they any less than (Jonny) Howson? Is (Rudy) Gestede any less than Assombalonga? Maybe tonight but that's just one night. They are still Championship players. It was our night on the penalties.
Woodgate, who made 10 changes to the team that lost to Brentford in the Championship on Saturday, said: "It was a terrible performance, not what I want from my teams but like I've said before we are thin on the ground.
"I told my players before the game if you don't turn up in any game you'll get done and we didn't deserve to win the game.
"I'll take a couple of pluses from Ashley Fletcher's performance second half, Paddy McNair, Nathan Wood – but apart from that, not good at all.
"We were really sloppy from the start, really, really sloppy. It was a chance for players to stake a claim in the team and they haven't done that."