The Clarets manager blamed referee Mark Cooper for his side's failure to come away with all three points, insisting the official would be more at home on Hackney Marshes.
Six bookings, a turned down penalty decision and a Richard Sadlier equaliser that Ternent saw as offside all contributed to the dour mood.
Three straight wins contrasted sharply with Millwall's four consecutive defeats and with only one defeat in fourteen Burnley still harbour genuine play-off ambitions, making the loss of two points all the harder to take.
Millwall boss Mark McGhee kept faith with the side humiliated by Portsmouth on Saturday, but by and large those same players spurned the opportunity to redeem themselves, falling short of the standards expected.
Their slow start allowed the visitors to settle and steal a lead in the 18th minute. Ian Moore, who had only just seen an effort beaten out by keeper Tony Warner made sure second time around, stabbing home after Lee Briscoe's initial effort had been saved.
Try as they might, Millwall could not muster a first-half response. They had chances, but looked every inch a side that had failed to register more than one goal in a game since January 18.
Steve Claridge headed over from six yards while Neil Harris should have capitalised on the otherwise superb Drissa Diallo's poor attempt at a clearance, but shot straight at Marlon Beresford.
Things hardly improved after the break for the home side, with Burnley content to sit on their lead until McGhee's triple substitution enlivened things.
David Livermore headed against the crossbar and with only seven minutes remaining the Lions deservedly levelled as Sadlier was on hand to poke home after Robbie Ryan had contrived to miss a golden opportunity.
The big Irishman should have sealed the points in time added on, but somehow Beresford pushed away the striker's header to ensure things ended even.