Rovers controlled the game for long periods and could have gone in front for the second time when Jason Roberts shot wide in the 81st minute with the goal at his mercy.
Less than a minute later Carlo Ancelotti’s side were heading towards all three points when the unmarked Ivanovic connected with a Yury Zhirkov cross at point-blank range.
Rovers boss Sam Allardyce will be left reflecting on how his side finished up with nothing after an exceptional first-half display capped by Benjani Mwaruwari’s first goal for the club.
Benjani headed a 21st-minute opener as Rovers pinned Chelsea back in their own half and ought to have moved further ahead before the break.
Instead they allowed Nicolas Anelka to fire home a 39th-minute equaliser and leave Chelsea no doubt relieved to have got off so lightly.
Starting the day five points clear at the top of the table, Ancelotti’s men had headed north seemingly only concerned by the continued absence of Frank Lampard due to injury.
But any sense of superiority which may have pervaded the visiting dressing room was soon swept away as struggling Rovers launched an all-out assault.
At the end of a first period notable for its one-sided action, Allardyce’s side could count themselves particularly unfortunate not to find themselves comfortably ahead.
Mame Biram Diouf and Brett Emerton were both denied shooting chances by the returning Alex before captain John Terry got a crucial block on a Morten Gamst Pedersen effort.
Moments later Mame Biram Diouf attempted an audacious lob over the back-pedalling Cech, who slipped but just managed to flap the ball out for a corner.
Benjani drilled a shot into Cech’s legs before he grabbed the richly-deserved opener, rising highest to nod home a cross from El-Hadji Diouf.
The visitors looked strangely flat with Michael Essien coming closest with a long-range effort on 37 minutes, but they only needed one chance to pull level.
The equaliser arrived in the 39th minute at the end of a flowing move which saw Florent Malouda cross from the left to Didier Drogba, who headed down for Anelka to sweep home.
To their credit Rovers were far from rattled and continue to create chances in the second half, mostly through the lively and energetic Mame Biram Diouf.
But the visitors fired another warning that it would only take a second to turn the game in their favour when Drogba crossed for Ashley Cole to poke a left-footed shot just wide.
And Chelsea came close again on the hour when Drogba’s cross from the right was stepped over by Malouda before Zhirkov drilled a low left-footed effort at Robinson.
Rovers should have regained the lead a minute later when Pedersen sent in a set-piece from the right which dribbled across goal before Mame Biram Diouf bundled it the wrong side of the post.
Moments after coming on as a substitute, Daniel Sturridge almost made an immediate impact when he ended a diagonal run with a crashing left-foot shot which flew into the side-netting.
But Rovers continued to press for a winner and it should have arrived in the 81st minute when Roberts turned inside Ashley Cole before shooting across Cech and wide of target.
From that moment it seemed inevitable Rovers would be punished and they duly were less than a minute later when Ivanovic nodded in Zhirkov’s left-wing cross at the far post.