A second half displayed described by coach Paul Alston as the "best this season" saw Luton hang on by a single point to register their eighth win of the season and sees them jump to 7th in the Midlands 2 East (South).
A depleted side looked to improved on recent performances and Luton gave a debut to centre and son of coach Alston, Harry.
A fine team performance was typified by Luton's John Kerr who had a standout game tackling ferociously. Playing a well drilled side who looked to break lose on many occasions the Newlands outfit stuck to their task and made the most of the few chances they created.
A first half dominated by Oakham saw them take only a 2 point lead into the break. Luton scrambled a try of their own to counter two by the men from Rutland. Harry Alston capped a fine debut by changing down a clearance close to the Oakham line before gathering and touching down. Hamm chipped in with a conversion and penalty.
The second period looked to set to be a repeat of the first as visitors re-established themselves with two penalties. But Luton fought back to within a point with a try by Davies and a brilliant touchline conversion by Hamm.
And then with mounting confidence Luton kicked a penalty into the corner. A clinical catch and drive to the line saw hooker Sheppard last man up as the pack crashed over to extend their lead to 4 points.
As time ran out Oakham grab a late penalty but a stalwart defensive display saw Luton hang on despite losing two players to injury.
First team manager Steve Evans said:"I'm incredibly proud of the effort from the team today, who demonstrated tenacity and determination to get this well needed win.
"We need to take forward the energy the boys displayed today into the rest of the season. We've shown what we are capable of, but now we need to consistency."