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.
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.
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.