What, exactly, is making you think that your system is "dragging"? Your board has SATA III ports, USB 3.0 ports, & while its PCIe x16 slot may "only" be PCIe v2.0, I can't think of any cards that will be throttled by that. And not that I have experience with the board in particular, but it certainly seems to be set up to facilitate overclocking (https://www.msi.com/Motherboard/760GA-P43-FX.html#hero-overview).
Will you see some performance gains by going to an Intel build? Yes, of course...as long as it's at least a Haswell Core i5/i7 or better. However, all too often in games (especially newer ones) you'll see the Intel system only get maybe a 10% leg up on the AMD systems, with the latter able to get at least 60FPS (given the right combination of GPU & resolution), which means you may not even notice the improvement. Whether or not you feel that's worth the expense to buy a new CPU, motherboard & RAM (since you'd most likely want a Skylake CPU, which means DDR memory) is up to you...but spending hundreds of dollars on those 3 items for a relatively small performance improvement is probably not a good idea.
That being said...there are some cost-effective steps you could take to pep up your system:
1. If you don't already have one, invest in an SSD for your primary disk. Actual game-performance won't increase (as you'll primarily see the changes when loading from the disk -- switching maps, initial load, etc.) but your system should see an improvement at the least in its boot-up & shut-down times. Best bet would be to at least get a 250GB SATA III model, & load your OS & most-used apps/games on it, with the other games & your data files kept on your old HDD...which, BTW, is more likely to be a reason that your system is "lagging" than the motherboard, so you may want to consider getting a replacement HDD as well.
2. Get a new GPU. Since you're using a GTX 960 right now, you're probably playing somewhere around 1080p resolutions (or even lower). At that resolution, an RX 480 or GTX 1060 is going to nearly double your average performance (http://www.techspot.com/review/1209-nvidia-geforce-gtx-1060/page10.html), either by letting you turn up the detail levels or simply letting you get closer to your monitor's refresh rate.