The veteran striker – the Lions' record goal scorer – took full advantage of some loose defending as the League Two visitors were outclassed in all departments.
After Chris Hackett's superb run from midfield created a chance for Gary Alexander to expertly volley home from 18-yards on 31, Harris poked home Millwall's second and his first seven minutes later.
Then Harris took over after the break with two stunning finishes in five second half minutes.
When a crisp one-two with Alexander set him clear after 68 minutes, the veteran striker guided the ball home and then he sent the home crowd into raptures with a stunning right-foot volley into the top corner.
The Lions dominated early, without creating any real openings, full back Alan Dunne setting the tone with a 20-yard drive after four minutes.
Burly midfielder Ali Fuseini came closest to opening the scoring in the 23rd minute when Jalal expertly tipped his long-range drive round the post.
Pacy winger Liam Feeney was the visitors' chief threat, creating a chance for former Lion Sammy Igoe, but Alan Dunne bravely blocked his snap-shot in the 16th minute.
Lee Bradbury – the striker filling in a right back – poked a difficult chance over the bar a minute after Harris' second after a long-throw but the Cherries did not look like scoring before the break.
Bournemouth forward Brett Pitman – last season's leading marksman – nearly pulled a goal back on 50 but his low shot curled a yard wide.
Jimmy Abdou should have added a third just after the hour, but the energetic midfielder somehow volleyed over from 6-yards.
However, Harris soon settled any lingering doubts of a Bournemouth comeback, securing the match-ball with real class.