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Despite Glen Wright scoring his first goal of the season to level the score midway through the first half, the visitors reclaimed their lead before half-time and netted two more goals in the second half.
Both teams started the match with attacking intent, and Sidmouth gained the early advantage in the 10th minute with a goal from Louis Spalding.
Wellington responded positively and nearly equalised in the 25th minute.
A well-placed corner found Wright, whose header was brilliantly saved by Sidmouth's goalkeeper, Elliot Driver.
However, Wright made the most of the subsequent corner, heading the ball over Driver and into the net to level the score at 1-1.
The Tangerines' progress was hampered by injuries to Isaac Baker and Wright in quick succession, both of whom had to leave the field.
Sidmouth capitalised on the disruption and regained the lead in the 41st minute, with a deflected ball finding its way past Wellington's defence and into the net.
The hosts continued to battle, creating a good opportunity in the 64th minute when Ben Evett raced through one-on-one with Driver, but the Sidmouth goalkeeper made a crucial save.
The Devonian visitors extended their advantage four minutes later with a header from Nathan Cooper following a corner.
Sidmouth added a fourth goal shortly after, with substitute Danny Pym scoring from around the penalty spot.
Wellington's injury woes worsened in the closing stages when Evett was forced to leave the field, leaving the team to complete the match with 10 men.