In a half where a series of incredible saves from James Belshaw denied the hosts, Bromley edged ahead through former Burnley man George Porter's accurate 12th minute finish.
Belshaw's opposite number was much busier in the second 45, but couldn't do anything about Knowles' tap in after Jack Emmett had teed up the striker to peg back The Ravens.
Simon Weaver had named the same starting XI that claimed a late point at Hartlepool on Tuesday night, with Kelvin Langmead partnering Callum Howe in central defence.
The contest started at quite a pace as George Thomson found himself through on goal in the opening 20 seconds, but saw his finish blocked by the alert David Gregory.
Town were made to rue their missed opportunity soon after as George Porter fired The Ravens into a 12th minute lead.
The former Burnley man was afforded space on the edge of the area and clinically fired into the bottom left corner.
An equaliser almost materialised four minutes later as George Thomson's corner found Howe, but the defender's header was cleared from the line.
Howe had picked up where he left off at Hartlepool on Tuesday night and got on the end of the resulting corner, only to see his effort once again hooked off the line, though some were convinced the ball had crossed the line.
With just over 20 minutes gone Porter was again afforded space on the edge of the area, but this time saw his shot towards the opposite corner bounce back across the goal via the post.
Bromley were creating chances at will, but couldn't find a way past James Belshaw for a second time as the stopper made a terrific triple save to first deny Omar Bugiel, then goalscorer Porter on two occasions.
Belshaw's next two saves were arguably more impressive as the Town number one twice denied the lively Porter, on the second occasion when the Bromley attacker looked certain to score!
Jack Emmett tried to craft an opening for Town before the half time whistle, though Bromley stood firm and were wondering how their lead was only one at the interval.
Belshaw was straight back into the thick of the action after the restart, keeping out Porter's shot that had taken a tricky deflection off Howe, though Gregory was forced to make a save at the other end not long after, keeping out Jack Muldoon's powerful shot.
Town almost cunjured up a leveller in the 54th minute as, after good work from Emmett and Dom Knowles, Williams edged in front of his marker and was unlucky not to squeeze his finish under the cross bar, but saw his attempt strike the frame of the goal.
Lloyd Kerry and Joe Leesley were introuduced in an effort to get Town back into the game, and the former was involved when Town's goal did come in the 73rd minute.
Kerry and Warren Burrell worked the ball well down the right before finding Emmett, who darted towards goal and found Knowles at the back post to tap in.
Belshaw's action packed afternoon was not quite done as the stopper tipped Jack Holland's flicked header round the post late on.
Emmett almost created a winning goal late on as he teed up Aaron Williams, but Gregory denied the striker at the near post and ensured Town claimed a point for a third successive fixture.
Harrogate Town: Belshaw, Burrell, Kitching, Falkingham, Howe, Burrell, Thomson (Kerry 66), Emmett, Knowles, Williams, Muldoon (Leesley 63)
Subs not used: Cracknell, Parker, Agnew
Goals: Knowles 73
Booked: Kerry, Leesley, Falkingham
Bromley: Gregory, Adebayo-Rowling, D Johnson, Sutherland, R Johnson (Okoye 57), Holland, Taymond, Bugiel, Meekums (Taylor 62), Porter, Ogedi-Uzokwe
Subs not used: Huxter, Taylor, Goodman, Enver
Goals: Porter 12
Referee: A Quelch