Yes. While Experiments reporting is session based, a user is kept into the test, seeing the same variation the whole time, so when they convert, they will be attributed to the test variation they have been seeing. This is limited to the same browser on the same device, so if the user changes devices (20% of the time for average eCommerce sites by my estimation) or clears cookies (1%) or changes browsers on the the same device (?), than Experiments (and nothing else legal) can track them.

Hope this helps.

PS(I suggest you look at other tools, as Experiments is the last tool I would recommend right now).

If they are using the same device, probably yes.

With Google Experiments, if they should see the same version of a page, and also be recorded as having used that version for getting credit for the transaction.

Google Experiments uses cookies. Now, if the user has some type of anti-cookie privacy software installed, is using a different browser, etc. good luck.

If they are using different devices, may be. For example they browsed on your site on their mobile, and then went back and used a desktop to purchase.

What Google has done, is upgrade Google Analytics to something called Universal Analytics, where Google attempts to follow users across platforms.

This above assumes, that the transaction tracking is set up correctly within Google Analytics (a big if, analytics set up can be tricky for many E-Commerce set up and will require help of a programmer).

Their are other programs, paid, which may be a better choice for doing Conversion Optimization. The two most popular ones are Visual Website Optimizer and optimizely.

And when doing testing, remember, document your tests and run them long enough so you get statistically significant results.

For example:

3 conversions on variable A, and 1 conversion on variable B over a holiday weekend is probably not representative enough to make changes to your site on. And even at 95%, that says there is a 1 in 20 chance your wrong! So run your test long enough, so for example, you are not getting only Saturday type traffic.

Good luck with your testing!

Google Experiments supports first click attribution, last click attribution, and even data-driven attribution. To specifically answer your question, will the variant that the user first saw get credit for the transaction or not depends on the attribution model selected .If the first click attribution model is used, the variant that the user initially saw during the experiment is likely to receive credit for the transaction.
However, it's important to note that Google Experiments allows you to define custom conversion goals and configure the attribution model within Google Analytics.