Watching from his home in Cornwall due to a positive coronavirus test, Neil Warnock saw Dominic Solanke strike the visitors into a 38th-minute lead with a neat back-heel.
But, despite surely bemoaning his side's lack of cutting edge, he will have been cheered by the sight of substitute Browne meeting a Paddy McNair cross to head a deserved equaliser 10 minutes from time.
Boro's failure to turn possession into clear-cut chances is one of the reasons why Warnock will have been pleased to tie up a three-year deal for former Arsenal striker Chupa Akpom, who was watching from the scarcely-populated stands.
His side dominated the opening exchanges, much to the joy of the socially-distanced Boro fans who had struck lucky in the ticket ballot and were allowed in as part of the EFL's pilot scheme.
They watched Djed Spence tee up George Saville, who dragged a low shot just wide in the third minute, then Marcus Tavernier drilled a low shot which was easily saved by Cherries keeper Asmir Begovic.
Boro were made to pay for their lack of cutting edge in the 38th minute when they switched off at Arnaut Danjuma's corner and Solanke nicked in to steer the ball beyond a helpless Marcus Bettinelli.
The hosts responded positively and should have been level two minutes into the second half when Grant Hall rose strongly to meet a corner from Tavernier, but he could only direct his header straight at a grateful Begovic in the Bournemouth goal.
But, for all their possession, Boro were still struggling to carve out clear-cut chances and most of the credit had to go to the solid Cherries centre of Dan Gosling and Colombian Jefferson Lerna.
Having cut a frustrated figure for much of the afternoon, Britt Assombalonga burst clear in the 65th minute, but was met on the edge of the box by the onrushing Begovic, who booted clear.
Saville was caught offside when he prodded home from an Assombalonga free-kick, and Boro were almost punished when Diego Rico fed Solanke on the counter-attack, and the striker was only denied a second by a superb block by Bettinelli with his feet.
But Boro finally grabbed the equaliser they deserved 10 minutes from time when McNair's ball in from the right evaded the Bournemouth back-line and fell kindly for the unmarked Browne to head home from point-blank range.
Boro were forced to cling on for their point with Solanke coming close again late on before Bettinelli reacted superbly to keep out Lerna's goal-bound header.
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