More on KentOnline
Welling began life in Vanarama National League South with an opening day loss at Oxford City on Saturday.
The new-look Wings, under manager Mark Goldberg, did not have enough attacking intent but lost out after conceding a penalty in either half.
Welling fell behind after conceding a 10th-minute penalty. Liam Enver-Marum sent keeper Chris Lewington the wrong way from the spot.
The Wings responded however and Danny Waldren just failed to connect with an Oliver Sprague free-kick.
However, there was no dispute about the quality of Waldren's 23rd-minute equaliser. City failed to deal with a free-kick and the loose ball was struck first time by the Welling midfielder from 20 yards beyond a stranded keeper.
Welling threatened through the tireless running of striker Harry Crawford but, with 10 debutants in their line-up, they failed to create many clear-cut chances.
Oxford regained the lead with a second spot-kick early in the second half.
Sprague failed to clear his lines and having missed his kick, clattered into a forward. Enver-Marum went the same way again to make it 2-1 after 57 minutes.
Welling introduced Warren Whitely, Jonte Smith and Bobby Devyne but they failed to have a shot at goal until the 87th minute when Whitely's near-post shot was turned behind.
Oxford used every trick in the book to wind the clock down, with keeper Jack Stevens the main culprit. He was eventually booked in stoppage time but he could have conceded a penalty himself when he wrestled Welling skipper Sean Francis to the ground moments earlier off the ball.
Welling: Lewington, Nelson, Brown, Corne, Francis, Leacock, Hatton, Waldren, Slabber, Crawford, Sprague. Subs: Whitely, Banton, Smith, Devyne, Johnson.
Attendance: 243.