The Gunners dominated the early exchanges and twice hit the woodwork, but Sonia Bompastor’s side weathered the pressure before growing into the contest.
The hosts broke the deadlock shortly before half-time after their high press forced Arsenal into trouble.
Wieke Kaptein won possession before Keira Walsh’s effort was parried by Anneke Borbe, allowing Thompson to collect the loose ball and fire home from close range.
The Gunners pushed for an equaliser after the restart, but Chelsea defended strongly, with Hannah Hampton helping preserve the clean sheet during a tense finish.
Victory moved the Blues within two points of Women’s Super League leaders Manchester City.
Chelsea now turn their attention to Wednesday’s Women’s Champions League trip to Lyon before visiting West Ham United in their next league fixture.