The West Londoners went into Saturday's Premier League fixture sitting in 11th position in the standings and boosted by Newcastle United beating seventh-placed West Ham United in a seven-goal thriller earlier in the day.
However, the Blues failed to take full advantage, despite going ahead just before half time and Burnley playing the entire second half with 10 men.
Lorenz Assignon was sent off for a second yellow card just before half time, Cole Palmer scoring the subsequent penalty, but Burnley were level two minutes after the restart through Josh Cullen.
Although Chelsea regained the lead during the latter stages through Palmer's second goal of the game, Burnley were always a threat and it came as little surprise when Dara O'Shea equalised three minutes later.
While Chelsea now find themselves within four points of seventh place with a game in hand, the result in isolation only increases the pressure on Pochettino's shoulders.
After the game at Stamford Bridge, Pochettino was unable to hide his frustration at dropping more points, admitting that his players must learn how to handle such situations that occur in games.
Speaking to Match Of The Day, Pochettino said: «So disappointed. We didn't show the consistency and the capacity to deal with and manage the game. We allowed them some actions from set pieces to get some possibility to believe.
»It is a result that shows that we need to improve in many areas if we want to be competitive.
«It was so easy for them to get to our last third. We were not aggressive enough in open play and defending set pieces. We need to
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