The depleted home side were looking to improve on recent performances and 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 Road outfit stuck to their task and made the most of the few chances they created.
Oakham dominated the first half and took a two point lead into the break, despite David Hamm opening the scoring with a penalty.
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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 chipping 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 to move 18-10 in front.
However, Luton fought back to within a point with a try by Chris Davies and a brilliant touchline conversion from Hamm.
Then with mounting confidence, Luton kicked a penalty into the corner as a clinical catch and drive to the line saw hooker Ryan Sheppard the last man up as the pack crashed over to extend their lead to four points at 22-18.
As time ran out Oakham grab a late penalty but a solid 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 as they demonstrated tenacity and determination to get this much-needed win.
“We need to take forward the energy the boys displayed into the rest of the season.
“We’ve shown what we are capable of, but now we need to consistency.”
Luton are without a league game this weekend.