A win for the Hammers will reduce the gap to second-placed Southampton to just two points with one game remaining, but they could only go into the break level with the Foxes.
The opening 20 minutes of the contest offered little in terms of goalmouth action, but the introduction of Lloyd Dyer gave the game an injection of spark.
At the other end Kevin Nolan's effort came close to breaking the deadlock, but it was Beckford who made amends for his earlier miss by opening the scoring.
Ben Marshall's smart play down the right saw him beat Paul Konchesky before delivering a precise cross for Beckford to head home.
Again though the Hammers responded well and were on level terms just five minutes later. Central defender Winston Reid was the unlikely scorer as he converted from Matt Taylor's low ball into the box.
The visitors nearly took a 2-1 lead into the break as in-form striker Ricardo Vaz Te struck goalwards, only for Wes Morgan to block well and deny the Portuguese forward.
Sam Allardyce's side require another goal without conceding to take the race for automatic promotion to the final day of the season.