The Rams looked in total control as they went into half-time with an 18-point advantage, as they looked for only their second win in the Championship this season.
Their tries came from Andy Kain, Robbie Ward, Paul Sykes and James Glover, with Sykes adding four goals.
But the Lancashire visitors turned the tables after the break, thanks to tries from Adam Neal, Michael Ward, Kieran Gill, George Tyson and Joe Burke. Scott Leatherbarrow kicked four goals.
Defeat leaves Dewsbury stranded on two points, only kept off bottom place by Bradford Bulls, who started the campaign with a 12-point deduction.
Dewsbury Rams: Guzdek, Morton, Crookes, Sykes, A. Brown, Speakman, Kain, Stringer, Ward, Teanby, T. Adamson, L. Adamson, A. Brown. Substitutes: Day, Douglas, Glover, Sheriffe.
Oldham: Turner, Clay, Tyson, Gill, Lepori, Leatherbarrow, Hewitt, Neal, Hughes, Burke, M. Wood, Langtree, Spencer. Substitutes: Owen, Ward, Bent, Gee.
Referee: T Crashley (RFL).
Bradford Bulls missed out on the chance to wipe out their 12-point penalty with a crushing 60-4 Championship defeat at French side Toulouse on Saturday.
The Odsal side have picked up five wins since the start of the Championship campaign, but never looked like collecting a sixth in France.
The Bulls trailed 24-4 at the break - their only scoring coming from a James Davies try - and it just got worse in the second half, shipping 12 tries in total.
Kuni Minga raced in for a hat-trick, while there were braces for Rhys Curran and Bastien Ader.
Other Toulouse tries came from Johnathan Ford, Gavin Marguette, Anthony Marion, Stan Robin and Bastien Canet. Tony Maurel kicked six goals to complete the scoring.
Toulouse: Maurel, Minga, Ader, Marguerite, Marcon, Ford, Robin, A. Bentley, Curran, Planas, Boyer, Marion, Canet. Substitutes: K. Bentley, Hepi, Leha, Mika.
Bradford Bulls: Mendeika, Oakes, Pryce, Kirk, Magrin, Larroyer, Caro, Campbell, Pickersgill, Hallas, Moore, Hodgson, Davies. Substitutes: Storton, Ryan, Wilkinson, Tomlinson.
Referee: C Campbell (RFL).