Boro appeared to be cruising to victory as they led 2-0 with 15 minutes remaining thanks to Charlie Taylor's first-half own goal and a fine strike by Samuel Silvera.
However, Isaac Price kickstarted a West Brom rally from a well-worked free-kick.
Just five minutes later the home side were level when Ged Wallace slotted in from Samuel Iling-Junior's cross.
A former member of Manchester United's backroom staff, Ramsay replaced the sacked Ryan Mason this week with the Baggies languishing in 18th in the table.
And they remain 10 points adrift of the play-off places after Burgzorg provided another late twist in the tale.
The Dutchman produced a cool finish back across goal into the bottom corner after a slick move involving Aidan Morris and Silvera.
Middlesbrough open up a five-point lead over third-placed Ipswich in the race for a place in the Premier League next season.