The 36-year-old was commanding at centre-back in his first PL appearance since featuring for Chelsea in December 2016, and went close to snatching the winning goal.
Hosts West Brom started brightly as Filip Krovinovic curled an effort narrowly wide and new signing Karlan Grant had a goal disallowed for offside.
Burnley's Ashley Barnes then forced a fine save from Sam Johnstone on 28 minutes.
Both sides traded chances in the second half, with Nick Pope superbly denying Grady Diangana and Ivanovic in quick succession on 62 minutes.
Moments later Chris Wood's header from Ashley Westwood's cross hit the crossbar, before Barnes was denied by Johnstone.
But neither side were able to claim their first win of the season, leaving West Brom 17th on two points and 18th-placed Burnley on one.
Budweiser King of the Match: Nick Pope
West Brom: 26 Oct v Brighton (A)
Burnley: 26 Oct v Spurs (H)