Westlake broke down Oldham's dogged resistance in the 95th minute as his side made it eight victories in their first nine games.
There were few chances in the Roses battle at Boundary Park as defences got on top to ensure a tense game which the Latics did not deserve to lose.
But Leeds are on a roll as they picked up their latest three points despite strikers Tresor Kandol and Jermaine Bedford both being suspended due to their red cards at Gillingham on Saturday.
Manager Dennis Wise handed debuts to new loan signings Mark De Vries and Wayne Andrews, who played for Oldham under Iain Dowie in season 2002-03.
They got little change from the Oldham back four, but Westlake did the job after being introduced from the bench just three minutes earlier.
Leeds shaded the goal-scoring opportunities, with the first coming on nine minutes when De Vries got away from Ryan Bertrand, but could only send his shot straight at goalkeeper Mark Crossley.
Oldham had a better chance after Craig Davies hit a superb diagonal ball to Andy Liddell, but Liddell lost possession by the box and Chris Taylor saw his shot from 18 yards cleared from near the goal-line.
Taylor also blasted a shot off target, before Andrews wasted the next opening for Leeds by shooting over just before half-time.
Leeds improved after the break and almost broke the deadlock through midfielder David Prutton, whose header from six yards was superbly kept out by Crossley.
The winner came from a cross by Sebastien Carole, which found Westlake at the far post for a first-time volley from ten yards.