The Cobblers finished with 10 men after having striker Roy O'Donovan sent off for a second yellow card in injury time when he jumped to prevent the taking of a throw-in.
In a game of few clear chances the visitors felt they were denied a 38th-minute penalty when Rovers substitute Matt Harrold tangled with Matt Heath.
Chris Hackett went close to an equaliser in the last minute of normal time with a curling free-kick that rebounded back off a post.
Northampton's best first-half chance fell to Clive Platt early on, but his close-range header was well save by Steve Mildenhall.
A closely-fought second half saw John-Joe O'Toole fire wide and Clarkson shoot into the side-netting for Rovers.
Aidy Boothroyd's side had six players booked by referee Mark Haywood, including midfielder Hackett for diving.
- Sports & Recreation
- David Clarkson
- Chris Hackett
- Matt Harrold