Hi,
first you can apply for open work permit (you don't need labor market opinion (LMO) for this kind of work permit and your work permit is not related to only one employer) as her's common-law partner and with that you can come here and work while she studies.
http://www.cic.gc.ca/english/study/work-spouse.asp
second, after she finishes her degree she can apply for post graduate work permit (duration is determined by duration of her studies)
http://www.cic.gc.ca/english/study/work-postgrad.asp
unfortunately, when your partner is on postgrad work permit you can't get again open work permit but you can apply for regular work permit (and that permit needs LMO)
http://www.cic.gc.ca/english/work/apply-who-eligible.asp
instead of this work permit, you could enroll to some one or two year university/college degree (as a student you can apply for a open work permit (after first six months of studies) and work during your studies) or enroll to some co-op program. If I was in your place I would choose this path instead of working while your partner is on postgrad work permit because I think it would benefit to your easier immigration and you would acquire canadian education which would help you to get a better job.
third, when your partner has accumulated one year of work experience with postgrad work permit she can apply for permanent residence under Canadian Experience Class (CEC) program
http://www.cic.gc.ca/english/immigrate/cec/apply-who.asp
After she secures permanent residence, you two can marry and she can sponsor your immigration as you are her family member (or you could apply under CEC program if you acquired enough of work experience and or studies).
http://www.cic.gc.ca/english/immigrate/sponsor/index.asp
There is no simple solution to your question but if both of you are patient and strong enough you can achieve all this in 3 or 4 years. Most important thing is that one of you gets permanent residence, after that, everything is much easier.
You could also check http://www.legit.ca/ and contact them. They might provide you will some specific help and/or advice for same-sex couples.
Best of luck to both of you
p.s. if I'm not mistaken together with work permits you will also need a visitor visa (I think that your partner also needed to get a visitor visa together with study visa)