Goals from Tommy Rowe and Josh Martin secured a 2-0 win for Rovers which sealed Crewe’s fate with four games remaining.
It was Crewe’s 15th defeat in 16 games and their 29th loss in total this season from 42 games.
A statement on Crewe’s website said: "The club would like to thank David for his substantial contribution, both as First Team Manager and in his previous role as Head of Academy.
“Alex Morris and Lee Bell have been appointed First Team Manager and Assistant Manager respectively, on an interim basis.
“Everybody at Crewe Alexandra would like to wish David every success in the future.”
Speaking following the loss to Rovers, Artell told the BBC that relegation had been "a long time in the making".
The 41-year-old said: "I think we've had six 17 and 18-year-old’s make their debuts [this season] and play a good number of games and, however it's come about, that's the situation we are in.
"By and large, the young kids have done ever so well, but across the board we haven't been good enough."