SegWit doesn't change the fee you need to pay in terms of sats per vbyte. You can make a 1 sat/vbyte transaction equally as well from a legacy address or a SegWit address, and both would have an equal chance of confirming in the next block. A SegWit and a legacy transaction with the same fee in terms of sats per vbyte will confirm at the same time (excluding the scenario of a backed up mempool with several megabytes of transactions at the same fee rate). Neither is given priority.

What SegWit does do is decrease the effective size of your transaction in terms of vbytes, so the same fee rate (in sats/vbyte) is cheaper overall. For example:

Let's say we have a legacy transaction which is 1000 vbytes, so 1 sat/vbytes costs 1000 sats.
The same transaction from a SegWit address might be only 500 vbytes. We can either pay 1 sat/vbyte, costing 500 sats and saving us 500 sats, or we can pay the same 1000 sats and end up confirming faster as we have paid 2 sats/vbyte. Or we can pay somewhere in between, and be both a little faster and a little cheaper.