Their fifth win in a row means that the M2ES league leaders remain unbeaten.
After falling behind to an early Oadby try and conversion, prop Dwayne Duffy opened the scoring for Luton touching down after good play by the forwards. In-form Rik Hobbs added the extras.
Luton took control of the game. Hobbs added another penalty before centre Chant was set free to score in the corner.
There was enough time for both sides to add a penalty before the half time whistle and Luton lead 18-10.
After the restart Luton appeared to take an unassailable lead. Hobbs added a brace of tries, first chasing down a loose ball and second forcing his way to the line again after good work by the pack.
But Oadby came back strong as they reminded their hosts why they are in contention for promotion and looked to punish Luton for poor defending.
Two tries and conversion brought Oadby to within a converted try of Luton. The home supporters endured a tense final 5 minutes but saw Luton hang on for a bonus point win.
First team manager Steve Evans said: “We knew it was going to be a challenge today because they are second in the league after us and Oadby Wyggs are everyone’s tip for promotion.
"I think if we are going to be taking anything away from this, it was a lack of discipline in the second half. You can never afford to do that in rugby."
Evans believed however that his side would hang onto victory despite witnessing his team advantage evaporate.
He joked: "It is not good for my heart this game. It's quite difficult to watch when you fear a significant lead go to waste. But that said, I got faith in this team as we had a strong defence."
Luton will now travel to play Oakham next Saturday.
Match Reporter
Fabian Solomon