Interim chairman Steff Wright said "We're very pleased and we think it's the right time to make the announcement.
"Keith has got a massive job to do in Cardiff and then in terms of putting his squad together for next season."
The manager, who has led his side to the League Two play-offs for the last three seasons, believes the Imps are a club who are going places.
Alexander said: "It wasn't a difficult decision to make. We can now push this club forward."