Luke Armstrong's first half brace made it three wins from three to start the season for Town, who claimed a first ever win against Leyton Orient.
Armstrong converted George Thomson's pull back, then bundled home at the far post after the midfielder's shot had been parried by Orient goalkeeper Lawrence Vigouroux.
Orient had the chance to pull a goal back from the penalty spot after the break, but Mark Oxley's save ensured it would be a miserable evening for the hosts and a successful first away trip for Town.
Weaver made just the one change from Saturday's win over Barrow as Lloyd Kerry made his first start of the season in place of Danilo Orsi, with Simon Power returning to the bench.
For Kenny Jackett's side, Tyrese Omotoye made his Orient debut, while Shadrach Ogie made his full EFL debut.
Town made the perfect start to the evening as Armstrong wrestled possession from the Orient backline, launching a quick Town attack and getting on the end of Thomson's pull back to open the scoring after just eight minutes, claiming our first goal at The Breyer Group Stadium in the process.
Orient responded by having a lot of the ball, but Town were working hard out of possession and still posing a threat at the other end, the lively Armstrong forcing a first Harrogate corner with a shot from a tight angle.
Pressure started to build from the hosts, with Omar Beckles firing across the face of goal, but Armstrong would double Town's lead just after the half hour mark.
Lewis Page had delivered from the left, with Thomson's shot proving too hot for Vigouroux to handle, the O's stopper's block being bundled in at the back post by Armstrong.
The hosts pressed for an equaliser before half time, but found Mark Oxley in good form, the Town stopper keeping out Harry Smith's downward header.
Orient were awarded a penalty six minutes after the restart as Referee Carl Brook judged Josh Falkingham to have stopped Dan Kemp's volley with his arm, but Oxley would again deny the hosts getting on the scoresheet, saving Smith's effort low to his right.
The hosts probed for a way back into the game, but were left frustrated by a Town side who continued to defend in numbers and threaten at the other end.
A first clean sheet of the season and another three points on the board send Town third in the early SkyBet League Two table, going into Saturday's clash with Exeter City.
Town: Oxley, Page (Fallowfield 85), Falkingham, Burrell, Thomson, Pattison, Kerry, Muldoon, Hall, McArdle, Armstrong
Subs not used: Smith, Martin, Orsi, Cracknell, Power
Goals: Armstrong 8, 31
Booked: Muldoon, Thirlwell
Leyton Orient: Vigouroux, James, Happe, Smith, Kemp, Archibold, Omotoye (Sotiriou 56), Pratley, Beckles, Ogie (Wood 56), Kyprianou
Subs not used: Clay, Papadopoulos, Sweeney, Nkrumah
Booked: Beckles, Archibold. Sotiriou
Referee: C Brook
Attendance: 3,942 (78 away)