The victory came despite India playing whole of the second half with 10 men after Sehnaj Singh was sent off for a dangerous tackle in the 41st minute in the Group D match which was played under constant drizzle and slippery conditions.
Robin Singh scored the all-important goal in the 10th minute at the Kanteerava Stadium to give India three points.
Despite the win, India still occupy the bottom of the five-team group with three points while Guam are just above them on seven points.
The group winners and four best runners-up teams will advance to the final round of qualifying for the 2018 FIFA World Cup. Asian giants Iran and Oman are on equal 18 points after six matches in the group.
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The next best 24 teams from the preliminary stage of the qualifiers will compete in a separate competition for the remaining 12 slots for the 24-team 2019 Asian Cup.
The home side suffered a jolt in the 41st minute after Sehnaj was shown a straight red card for a late and high tackle on Justin Lee. But Stephen Constantine's side hung onto the lead despite incessant attack from Guam in the second half.
Goalkeeper Gurpreet Singh Sandhu also did a great job under the bar by making numerous saves late in the match after Guam went all out looking for an equaliser.