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Two first-half goals moved Welling up to fourth in National League South on Saturday.
The Wings proved too strong for East Thurrock with Brendan Kiernan and Jack Jebb on target.
It was a third win in four games for Steve King's side, who are now just four points behind the leaders.
Despite their lowly position, East Thurrock certainly gave Welling plenty to think about early on.
The impressive Tom Derry headed against the crossbar as early as the second minute before Sam Higgins' deflected free-kick went behind for a corner and Wings keeper Dan Wilks was out smartly to deny Roman Michael-Percil.
Welling went ahead, however, on 17 minutes when Jebb's corner was brilliantly volleyed home by Kiernan.
The hosts added a second 10 minutes later when Jebb cleared the ball from inside his own box, Danny Mills showed great desire and ability to win possession on halfway and the move ended with Craig Braham-Barrett setting up Jebb to fire in from the edge of the box.
Bradley Goldberg was then denied by keeper Jack Giddens while Michael-Percil was booked at the other end by referee Robert Massey-Ellis for diving as he tried to win a spot-kick.
Welling were awarded a spot-kick six minutes before the break when a quick corner saw Kiernan fouled in the box by Tom Stephen. However, Goldberg's run-up telegraphed his intentions and Giddens was down to make a comfortable save.
East Thurrock thought they had a goal back on 64 minutes but keeper Wilks produced an outstanding one-handed stop to claw away Michael-Percil's 20-yard curler which seemed destined for the top corner.
Welling then head three chances in a minute at the other end but Goldberg, Mills and Kiernan were all denied by Giddens.
Mills headed wide late on but East Thurrock ran out of steam and Welling were able to make it back-to-back clean sheets.
Welling: Wilks, Hurst, Braham-Barrett, Ijaha, Ambroisine, Audel, L'Ghoul, Jebb (Paul-Kissock 62mins), Mills, Goldberg (McCallum 77mins), Kiernan (Mendy 84mins). Subs not used: Mendes, McDonald.
Attendance: 547.