Though I'm still not a great fan of powered monitor speakers when the possibility of mixing/matching pre/power amplification for passive speakers is an option, if you're "upgrading" from one of those 2.0 or 2.1 "mini-speaker/sub" computer speaker systems, just about anything will be an improvement. Improvement is always good, and in a case like this, you can improve the system further, one bit at a time, as inspiration and/or budget allows.
For years, I used a Creative Labs "2.1 speaker system in a box" for my computers (sounded better than many speakers that come stock with some computers), but more recently, my computer's sound system has undergone two major upgrades (two complete system changes, in fact), and finally, it can produce a pretty decent sound for near field "desktop" listening (and even beyond, for a small to moderate sized room). You can see my current computer sound system listed in my profile.
During this recent two part upgrade process, I did switch from using the computer's analog output to using digital USB output to an external DAC (Schiit Bifrost). This was certainly a major improvement; all other components remaining the same during the initial period of that particular switch, so the improvement from this single change was obvious).
By the way, my "secondary" portable headphone DAC/amp is an Audioengine D1 ("primary" being a hybrid tube/ss HiFiMan EF2A DAC/headphone amp). I prefer the sound I get from the EF2A (portable headphone usage only), but if it's a concern, it's a bit bulkier than the Audioengine D1 (unit itself is bulkier, and it requires a fairly bulky wall wart power supply as well). The Audioengine D1 is very tiny and doesn't require an external power supply, as it gets all its power via the USB connection.
For a short time, while the Bifrost DAC was back at Schiit for some repairs, I used the Audioengine D1's DAC to feed the desktop system's integrated amp, as it was still a definite improvement over the computer's analog output (the pair of stereo analog RCA outputs on the D1 is a convenient feature of this little DAC/headphone amp). While I do like the D1 "for what it is", if you eventually want to get a DAC for your desktop system, I feel you can do better (and it would be well worth the patience if you had to wait a bit longer in order to purchase a more expensive "full size" DAC).
So, to get back to your original question: Yes, I believe that even just using the computer's analog output for now, the Audioengine A2 speakers would represent a definite improvement over stock or lesser aftermarket "mini speaker systems". Then, if you decide to improve other components (external DAC, especially, and perhaps add a good subwoofer as well), your sound could further improve, thereby taking even greater advantage of the A2 speaker upgrade.
Edits: 08/03/12
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