A sweetly-struck Marcus Tavernier opener would force the Hatters to do it the hard way, but Rob Edwards's men rose to the challenge and dramatically fought back through Jordan Clark and Morris to move level on points with 17th-placed Nottingham Forest.
A rare piece of good news arrived for Luton on the fitness front, as Reece Burke was given the green light to return to the hosts' backline, while Justin Kluivert and Antoine Semenyo were promoted back to the Bournemouth XI after combining for the winning goal against Crystal Palace in midweek.
Despite a positive start from the hosts, they survived a huge scare in the 11th minute, as Tavernier's well-hit free kick struck the woodwork with Thomas Kaminski scrambling.
Andoni Iraola's men subsequently took the game by the scruff of the neck, as Tavernier sent another curler wide in the 13th minute, four moments before Kluivert also missed the mark as a beleaguered Luton - leaving plenty of space in behind - clung on.
A few days on from diverting the ball into his own net against Arsenal, Hatters defender Daiki Hashioka nearly suffered the same fate in the 25th minute during a goalmouth scramble from a corner, but Kaminski managed to smother close to his goal-line.
Profligacy from both sides somehow kept the tie goalless heading into half time, and as was the case at the start of the opening 45, Luton turned the screw early on in the second period, as Morris tested Neto from a tight angle in the 47th minute.
The Brazilian goalkeeper repeated his trick from the Luton man with an even better save in the 51st minute, but as the hosts failed to make the most of their early pressure, Iraola's men did, as Tavernier was invited to shoot from distance and found the far corner with a brilliant low strike in minute 52.
The Cherries were still conceding plenty of chances to their hosts after taking the lead, as only the post denied Morris an equaliser in the 64th minute, before Adam Smith pivotally denied Clark on the rebound.
However, the latter was not to be denied nine minutes later, as the ball broke kindly for Clark when Luke Berry and Philip Billing both stuck a foot in to regain possession, and the Hatters man capitalised with a fine finish into the bottom corner.
Edwards's men were not content to declare at one, and just before the Kenilworth Road scrap entered additional time, the hosts dramatically took the lead, as Morris escaped the attention of Ilya Zabarnyi to meet Cauley Woodrow's delicious delivery with a deft side-footed volley.
The hosts' pulsating triumph was not enough to lift them above the dotted line, but they are only behind Nottingham Forest on goal difference having played a game more, while the Cherries stay safely in mid-table in 12th.
Luton now face the toughest possible test next weekend - travelling to the Etihad to face Manchester City on April 13 - while Bournemouth coincidentally host the champions' bitter rivals Manchester United in seven days' time.