Amond’s sixth-minute opener was cancelled out by Exeter’s Jayden Stockley early in the second half.
But it was the Exiles who came out on top as Willmott fired home from 25 yards in the 66th minute.
Newport were ahead early on. Matty Dolan’s corner was struck against the Exeter post by home skipper Joss Labadie before Amond’s glancing header from Willmott’s inch-perfect cross opened the scoring.
Amond had a shot blocked before the break and Exeter’s Lloyd James was lucky to escape with just a booking following a lunge on Mickey Demetriou.
Newport defender Dan Butler cleared James’ close-range effort off the line at the start of the second half and it wasn’t long before Exeter were level as Jake Taylor’s cross was headed home by Stockley.
James then missed with a sitter of a header from six yards out and it proved costly as, out of nowhere, Newport went back ahead thanks to a stunning Willmott strike from the edge of the box.
Amond hit the side netting but Willmott’s moment of magic proved enough for Newport to somehow hang on despite eight minutes of added time.
Northampton striker Kevin van Veen is hoping to make his first start in over a month in the home game against Shrewsbury.
Derby's decision to postpone Sunday's home Sky Bet Championship game against Cardiff was made independently, Derbyshire Police have confirmed.
Rochdale have no new injury worries ahead of Tuesday's Sky Bet League One relegation battle with fellow strugglers Fleetwood.