Marcus Tavernier gave Bournemouth the lead early in the second half with a crisp shot from just outside the corner of the penalty area.
But Jordan Clark restored parity with a clinical 73rd-minute strike from inside the box.
The Hatters ended a run of 10 games without a win in dramatic fashion when Morris guided the ball home from Cauley Woodrow's cross in the 90th minute.
Luton remain in the drop zone, but only behind 17th-placed Nottingham Forest, who are away to Tottenham on Sunday, on goal difference.
More to follow.
How to play Rate players out of 10 throughout or after the game. The rater will close 30 minutes after the final whistle.
Rating range key1 = Give it up10 = Pure perfection
Luton Town Starting XI
Squad number24Player nameKaminski
Average rating 6.29
Squad number12Player nameKabore
Average rating 6.73
Squad number15Player nameMengi
Average rating 6.78
Squad number16Player nameBurke
Average rating 6.59
Squad number27Player nameHashioka
Average rating 6.32
Squad number6Player nameBarkley
Average rating 7.11
Squad number18Player nameClark
Average rating 6.88
Squad number45Player nameDoughty
Average rating 6.75
Squad number30Player nameTownsend
Average rating 6.60
Squad number14Player nameChong
Average rating 6.61
Squad number9Player nameMorris
Average rating 7.05
Squad number8Player nameBerry
Average rating 7.06
Squad number10Player nameWoodrow
Average rating 7.52
Squad number32Player nameOnyedinma
Average rating 6.92
AFC Bournemouth Starting XI
Squad number1Player nameNeto
Average rating 6.63
Squad number15Player nameA Smith
Average rating 6.34
Squad number27Player nameZabarnyi
Average rating 6.70
Squad number5Player nameKelly
Average rating 6.47
Squad number3Player nameKerkez
Average rating 6.68
Squad number4Player nameL Cook
Average rating 6.62
Squad number10Player nameChristie
Average rating 7.47
Squad number16Player nameTavernier
Average rating 6.88
Squad number19Player nameKluivert
Average rating 6.86
Squad number24Player nameSemenyo
Average rating 6.96
Squad number9Player nameSolanke
Average rating 7.27
Squad number11Player nameOuattara
Average rating 6.18
Squad number14Player nameScott
Average rating 5.36
Squad number25Player nameSenesi
Average rating 4.84
Squad number26Player nameUnal
Average rating 6.30
Squad number29Player nameBilling
Average rating 6.45
Line-ups Luton Formation 3-4-2-1
12Kabore15Mengi16Burke
27Hashioka6Barkley18Clark45Doughty
30Townsend14Chong
24Kaminski 12Kabore 15Mengi 16Burke 27HashiokaSubstituted forBerryat 62'minutes 6Barkley 18Clark 45Doughty 30TownsendSubstituted forOnyedinmaat 62'minutes 14ChongSubstituted forWoodrowat 80'minutes 9Morris Substitutes 1Shea 8Berry 10Woodrow 17Mpanzu 23Krul 32Onyedinma 38Johnson 43Nelson 44Piesold Bournemouth Formation 4-2-3-1
15A Smith27Zabarnyi5Kelly3Kerkez
4L Cook10Christie
16Tavernier19Kluivert24Semenyo
1Neto 15A Smith 27Zabarnyi 5Kelly 3KerkezSubstituted forSenesiat 86'minutes 4L CookSubstituted forScottat 87'minutes 10Christie 16TavernierSubstituted forOuattaraat 59'minutes 19KluivertSubstituted forBillingat 68'minutes 24SemenyoSubstituted forUnalat 69'minutes 9Solanke Substitutes 8Faivre 11Ouattara 14Scott 23Hill 25Senesi 26Unal 29Billing 37Aarons 42Travers Referee: Andy Madley Attendance: 11,797 Match Stats Live Text Post update