Northampton dominated the opening stages and Day crossed the line in the fourth minute to start the scoring.
Steven Myler, who converted Day's try, added six more points from two penalties when Newcastle twice came in offside.
Mike Delany kicked three penalties through the posts before half time, before Myler took the half-time score to 16-9 in Saints' favour.
Todd Clever was sin-binned in the 51st minute, and Day made Newcastle pay three minutes later when he touched down after a driving maul.
Luther Burrell added another try just five minutes later, and Foden secured the bonus point after consultation from the TMO.
Foden added his second, and Northampton's fifth, in the 71st minute after a scrum on the five-yard line.