Jay Tabb scored twice as Coventry comfortably beat Southampton at the Ricoh Arena.
The home side went in front as Tabb powerfully headed home, before Elliot Ward's chipped penalty was comfortably saved by Southampton's Kelvin Davis.
However, Coventry did double their lead as McKenzie swept home from close range before Leon Best tapped in a third.
Andrew Surman pulled one back for the visitors, but Tabb's half-volley late on put the result beyond doubt.
Coventry manager Chris Coleman:
"We could have been four up at half-time - I don't think anybody could have had too many complaints if we had been.
"We had a great start to the second half but then Southampton changed their shape and put us under a little bit of pressure, but it would have been a travesty had we got anything else but a win.
"We created a lot of chances and scored four goals so that's a great result for us."
Southampton head coach Jan Poortvliet:
"We have to learn from this because we played two good games and then we perform like that.
"Maybe it was over-confidence and we thought that we could do anything with the ball so we did not fight for it.
"Today we played as individuals rather than a team - and we need to learn to be switched on, concentrating and motivated for a full 90 minutes."
Coventry: Westwood, Osbourne, Ward, Dann, Fox (Hall 81), Tabb, Gunnarsson, Doyle, Best (Morrison 81), Eastwood (Mifsud 89), McKenzie.
Subs Not Used: Marshall, Beuzelin.
Booked: Best.
Goals: Tabb 19, McKenzie 33, Best 47, Tabb 87.
Southampton: Davis, James (Gillett 58), Perry, Lancashire, Mills, Cork, Wotton (Wright-Phillips 58), McGoldrick, Lallana (Pekhart 59), Surman, Robertson.
Subs Not Used: Bialkowski, White.
Booked: Davis, James.
Goals: Surman 63.
Att: 15,518
Ref: Michael Jones (Cheshire).
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