Ben Roberts put the College in front with a try on eight minutes after Hartpury jacked back from a driving lineout, with Burns slotting the extras from the touchline.
Burns and the visitors’ Oscar Thomas exchanged penalties before Hartpury’s James Gibbons touched down from a catch-and-drive for the host’s second try.
Fullback Thomas and fly-half Burns’ kicking duel continued as the former kicked two to the latter’s one, leaving the college side holding a nine-point lead at half time.
The second period began sloppily for Hartpury when they gifted Irish a scrum for being ahead of the ball at the kickoff.
But it proved to be a blessing in disguise as the home pack won possession against the head and Burns ghosted through the off-guard Wild Geese defence to run home unopposed for seven points.
The hosts could have extended their advantage as Steve Leonard and Terry Fanolua went close in quick succession, with Hartpury seemingly set to seize control of proceedings.
But it was their opponents who grabbed the next score as scrum-half Tom Eastham sneaked over on the blindside seconds after replacing Ed Hoadley, improved by Thomas.
Thomas swiftly brought Irish to within two points when he converted Andrew Evans’ five-pointer, the lock capitalising on a lost Hartpury lineout on their own five-metre line.
The home side steadied the ship in the final quarter-of-an-hour, with two penalties from Burns taking them out of range.
There was, however, still time for Irish to claim a deserved losing bonus point when Thomas booted his fourth penalty with the last kick of the game.
Hartpury would like to thank London Irish Wild Geese for an excellent match and look forward to the return fixture in April.
Hartpury’s next match is away at Ampthill on Saturday 14th December.
Final score: Hartpury College 31-26 London Irish Wild Geese
Tries: Roberts, Gibbons, Burns
Conversions: Burns (2)
Penalties: Burns (4)
Gareth Thompson; Reuben Haile (Mark Harrison 16), Steve Leonard, Terry Fanolua, Ben Roberts; Billy Burns, Grant Hancox (Mitch Spencer 66); James Gibbons, Chris Fortey, Wayne Thompson (Alec Clarey 64); Dave McKee, Jonny Hill (Tom MacDonald 56); Andy Chesters, Alex Woodburn, Lewis Ludlow. Not used: Nathan Taylor