County were thoroughly outclassed in the first half but Davies' 30-yard screamer and Smith's header sent the League One side into the third round for the first time in four years despite Troy Deeney's 78th-minute goal.
Watford flew out of the traps and only the post denied Stephen McGinn from breaking the deadlock with a header in the fourth minute.
Magpies keeper Stuart Nelson had to be at his best to deny long-range efforts from Will Buckley and Marvin Sordell, while Sordell spurned a gilt-edged chance to put the Hornets in front after 19 minutes when he dallied in putting the ball into an empty net after rounding Nelson and was superbly tackled by John Thompson.
Following a lull in action, McGinn forced another fine save from Stuart Nelson on the stroke of half-time.
The hosts were made to pay for their failure to convert one of their several first-half chances when Notts County opened the scoring just seconds after the interval.
After Deeney had fouled Smith, Davies' superb 30-yard free-kick whistled into the bottom corner of the net to leave the home fans stunned.
Galvanised by the goal, Craig Short's men were no longer on the back-foot and though Nelson again denied Sordell, Smith doubled his side's lead in the 67th minute.
Shortly after coming off the bench, Craig Westcarr's chip into the box was headed in by the County striker.
However, Deeney halved the deficit with a goal on his debut 12 minutes from time when he headed in from Danny Graham's cross.
Graham missed the target from close range and Sordell was again foiled by Nelson as Notts County held on for the win.
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