They won the toss and batted first on a slow, turning wicket, posting 134-4 with good contributions from Adam Phippen (48), Mike Higgs (41) and Will Taylor (34no).
When Cumnor replied, they lost wickets regularly and were bowled out for 104 in the 19th over.
There were two wickets each for Higgs, Phippen, Chris Jeffcock and George Patey.
Oxford await Thame in the semi-final.
Meanwhile, in the Home Counties league last Saturday, Thame produced a much-improved display to force a draw with their local rivals Horspath at Church Meadow.
Horspath won the toss and elected to bat and good opening spells from Rooney and Lachlan helped the home side off to a strong start, reducing their visitors to 30-3.
However, 60 from Ward and further contributions by Ali and Foster took Horspath up to a challenging 195-9.
In reply, Thame struggled early on and poor shots and a couple of run outs left them reeling at 40-5.
However. a half-century from McDonald and contributions from Higgs and Phippen meant Thame got to 154-8 at the end of 54 overs.
Town travel to Cookham Dean this Saturday for a crucial match, with renewed heart.
Thame IIs travelled to second-in-the-table Abingdon Vale IIs knowing that nothing less than victory would suffice to keep their title-challenge on track, and they did not disappoint.
The home side won the toss and batted but only Luke Charlesworth (69) looked comfortable as Ali Thomas (three for 54) took key wickets, supported well by Ian McStay (3-41), as Abingdon could only manage a below-par 171-9 off their 53 overs.
After a good start by Richard Carr (18) and Jimmy Nielsen (32), Thame scored at almost four-an-over and, with Matt Furness in charge (60no), were comfortable winners by five wickets, from 44.3 overs.
The IIs host Kingston Bagpuize firsts this Saturday at Church Meadow.
Thame IIIsI took on Dorchester-on-Thames at Church Meadow on Sunday in a 40-over friendly to get some much-needed practice before their crunch match away at Buckingham this Saturday.
Dorchester won the toss and elected to bat with boundaries coming freely in the opening stand, but Matt Anderson’s 2-28 pegged them back.
The visitors nevertheless posted an imposing 221-4, with J. Neal scoring 79 not out.
In reply, Thame set off at a gallop as Danny Thompson opened up with 23. It was very much Christian Warwick’s day though as he took the attack to the bowling and hit his maiden century, ending on an impressive 132 not out.
With support from Evan Davies (35) and Charlie Greenhough (13no) Thame reached the target off only 27 overs.