Britain scored their first two goals through penalty corners from Glen Kirkham (28th) and Ben Hawes (43rd).
India struck back, scoring twice in the space of four minutes through a penalty corner conversion from Vokkaliga Raghunath (55th) and a stunning deflection by forward S.V. Sunil (59th) to bounce back in style.
But three minutes after drawing parity, a poor defensive lapse cost India the match as three defenders missed a through ball that setup Catlin to score the winner for Britain.
The win helped Britain moved to the second position with four points from two matches behind table-toppers New Zealand, who registered their third successive victory by defeating Pakistan 3-1 earlier in the day.
India made a good start against Britian as their speed caught the rival defence completely off-guard. India came close to open the scoring as early as in the third minute but Sunil failed to connect to a precise Tushar Khandkar's cross from the right.
Britain were lucky to break the deadlock in the 28th minute from their third penalty corner. India goalkeeper PR Sreejesh made a brilliant block to stop Ashley Jackson's shot but Kirkham slammed in from the rebound to give his side the lead. Hawes converted a set-piece penalty corner to double the lead in the 43rd minute.
Kiwis stay on top
New Zealand posted their third successive victory by prevailing 3-1 over an erratic Pakistan. By virtue of this win, New Zealand not only boosted their points tally to nine but also consolidated their position on top of the standings.
After taking a three-goal lead four minutes into the second session, the Black Sticks had to ward off numerous Pakistani raids from which the Asian champions could score only once. New Zealand had earlier outplayed India and Argentina in their previous matches.