Richmond's failure to award United a penalty incensed Craig Levein's players and drove them on to equalise before almost snatching a winner.
It had been a tepid opening by both sides, with the hosts creating their first chance in the 11th minute with a well-worked three-man move ending with David Robertson's 12-yard header going into the top corner only for Artur Boruc to tip over.
And nine minutes later with little to trouble either keeper, it was United again who came close with a Danny Grainger 16-yard volley skimming just over the bar.
Despite all their possession, there had been little from the Celts in the opening half to suggest title-winning form.
Their first sight of goal came in the 34th minute with a Marc Luque Crosas cross being headed over from eight yards by Jan Vennegoor of Hesselink.
With the half-time whistle drawing nearer, Aidan McGeady almost caught out Lukasz Zaluska with a wicked left-foot shot from 20 yards out which the keeper did well to turn round for a corner.
Six minutes after the break, the Hoops grabbed the lead with a fast-flowing move ending with Paul Hartley's shot taking a fortuitous deflection off a United defender.
Ex-Arab Mark Wilson began the move, passing to McGeady whose deep ball was knocked back by Scott Brown for the midfielder to fire in from 15 yards.
Two minutes later, United were denied a penalty which was to change the game.
Wilson chopped down Roy O'Donovan but, amazingly, Richmond waved away the entire United team's claim.
Fortunately for him it was his last decision as he was replaced by the fourth official Iain Brines.
But Celtic recovered from their break with a fantastic Shunsuke Nakamura volley from 25 yards being matched by the save from Zaluska.
And as the game heated up, Warren Feeney thundered a volley straight into the arms of Boruc.
With ten minutes remaining, the Terrors clawed back their deserved equaliser with a great goal from Francisco Sandaza.
A magnificent pass on the touchline form Grainger sent Craig Conway on his way and with his first touch he found the ex-Valencia striker, who glided the ball over Boruc and high into the net.