The Irons were pegged back just before the half-time whistle when Johann Gudmundsson equalised with a goal against the run of play.
Chris Wood levelled the tie-up once again ten minutes later before the Brazil global added his second of the game five minutes from time.
It was nearly 2-0 to the hosts when Anderson's goal-bound effort was miraculously kept out by Ben Mee, while the lively Grady Diangana should have been awarded a penalty after a trip from Steven Defour. However, the Brazilian made amends shortly after when his shot took a huge deflection off Mee to leave Hart with no chance.
Anderson restored the hosts' lead in the 68th minute when latching on to Diangana's throughball, only for Wood to head in a Robbie Brady corner and level things up soon after.
Having opened the scoring with his first goal of the season 16 minutes in, Son, one of 10 changes and making his 150th Spurs appearance, doubled his tally with a composed finish early in the second half to set Tottenham on their way to a second victory at London Stadium in the space of 12 days.
And Hernandez - introduced for Pedro Obiang in the 61st minute - put the game to bed at the end after racing on to Michail Antonio's pass. He's done a lot for me so far. They were not great defensively, however.
Diangana has had opportunities in the last few games in the West Ham side to impress due to a string of injuries, but he surely now has to be a real contender to become a regular starter under Pellegrini. He probably should have had a penalty as well.
While the competition represents the best chance for Spurs to win their first trophy since the 2008 League Cup, Pochettino says he is putting player health ahead of the pursuit of silverware.
After the trip to West Ham, Tottenham return to league action at Wolverhampton Wanderers on Saturday.
Burnley have now lost their last three matches, shipping 13 goals in the process.
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