The former Bolton Wanderers and Blackburn Rovers manager enjoyed a productive return to the north west on the eve of his 60th birthday as the visitors came away from Turf Moor with maximum points.
After coming under the pressure for much of the first half, West Ham took the lead shortly after the interval through Diafra Sakho's header - the striker extending his goalscoring run to six successive matches in all competitions.
That lead was doubled soon afterwards by Enner Valencia and although George Boyd pulled one back for Burnley, substitute Carlton Cole restored the two-goal advantage with 20 minutes remaining.
"I said to the players before the game 'come on lads we need a result today because it's my birthday and if I can't get a result it's not going to be very pleasing for me when I'm with my old mates'," he said.
Both Sakho and Valencia's goals came from superb crosses from wide positions, and Allardyce revealed a change in tactics at half-time was the reason behind their fourth Premier League success of the season.
"I think a change of system was the key element for us winning the game, which then brought better outstanding quality from us win wide areas," he said.
"It [the first-half display] wasn't good enough. The passing was lethargic, it wasn't crisp or sharp enough and we weren't getting the forward balls with quality quick enough, which allowed them to get back behind the ball.
"Our system was not working as well as their system so we changed to 4-2-3-1 and told the full-backs to push on and start getting in the game more and more.
"The providers of both our first two goals were outstanding crosses from Aaron Cresswell and Carl Jenkinson.
"Then when Carlton comes on and scored the third after we've gifted them a goal to get them back in that just summed up our performance today and I think from our point of view it's our first back-to-back victories."
