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Welling were beaten by a goal in either half against play-off chasing Alfreton on Saturday.
The Wings, who endured a 12-hour round trip to Gateshead on Tuesday night, came up against a physical and robust outfit closing in on the play-offs who needed a second-half penalty to claim the points.
Alfreton's approach made few friends at Park View Road but they also added a touch of quality when required as the game effectively turned on two incidents in either half.
Welling were denied a penalty when Harry Beautyman was clearly pushed in the back as referee Wayne Barratt - who sent-off Blaine Hudson in the 3-1 defeat at Aldershot in December - did not answer their pleas.
Play instantly switched to the other end where Shaun Harrad's backheel set up Josh Law, who beat keeper Lee Butcher at the near post.
Alfreton then dominated possession but Welling defended well and were level just before half-time when Ross Lafayette crossed for Kurtis Guthrie to prod home at the near post.
Lafayette then went clean through early in the second half but, having lifted the ball over keeper Jon Worsnop, he also lifted it wide of the target.
Welling paid the price for that miss within a minute as Connor McLaren fouled John Akinde, when he was going away from goal and Harrad tucked away the spot-kick.
Former Ebbsfleet striker Akinde was then fortunate not to be sent-off for an elbow on Doug Bergqvist as, already on a yellow card, not even a foul was spotted by the officials. Not surprisingly, Alfreton substituted Akinde shortly after.
Welling introduced loan signing Harry Cormick for the final half-hour and the Bournemouth youngster nearly had an immediate impact, firing at Worsnop from long range.
The hosts piled forward late on despite a torrential storm which included thunder, hail and lightning five minutes from time. Referee Barratt amazingly kept the players on the pitch instead of coming off to wait for the worst of the weather to pass.
Welling nearly snatched an equaliser during the five minutes of stoppages but Joe Healy shot wide when well placed inside the box.
Video: Tom Riding / WingsTV