Some of the White Hart Lane faithful were still taking their seats when Arsenal took the lead, with Rosicky leaving Hugo Lloris grasping at thin air with a venomous right-foot strike from the corner of the area which flew across the goalkeeper and into the back of the net.
Rosicky had picked up possession in his own half before advancing into the space vacated by Danny Rose's forward foray before playing a slightly fortuitous one-two with Alex Oxlade-Chamberlain and then finding the top corner.
The Gunners were forced onto the back foot for long periods of the north London derby, but the closest Spurs came to levelling was in the second period when Wojciech Szczesny inexplicably failed to gather a routine cross from the right and had Laurent Koscielny to thank for denying Nacer Chadli a certain leveller.
Arsenal are now third, only behind Liverpool on goal difference and just four points adrift of Chelsea and with a game in hand on the leaders, while they now boast a nine-point cushion over their derby rivals in fifth.
Arsenal were ahead inside two minutes of Arsene Wenger's 999th match in charge with a goal of the highest order.
Sherwood's side - whose collective character had been questioned by the manager - soon regrouped and caused Arsenal a few problems with probing balls into the penalty area.
Arsenal, though, retained possession well and it should have been 2-0 on 15 minutes when Oxlade-Chamberlain robbed Nabil Bentaleb on the halfway line and sprinted clear.
The England man darted to the edge of the penalty area, but as Lloris came off his line the Arsenal man screwed his attempted chip horribly wide - with Lukas Podolski screaming for a square pass for what would have been a tap-in.
That let-off gave a huge lift to the home side, who immediately went on the offensive with a low cross from Andros Townsend on the right flying through the six-yard box.
On 21 minutes, Arsenal full-back Bacary Sagna slid into a challenge with Danny Rose which - although he probably touched the ball first - totally upended the Spurs defender. Sherwood was off the bench demanding a red card but, after a stern talking-to, referee Mike Dean produced only a yellow.
Spurs continued to play a very high defensive line, which was breached again when Podolski raced clear down the left. His cut-back was just behind Rosicky but Oxlade-Chamberlain connected with a strike that was deflected out for a corner.
The home side should have been level at the start of the second half following two terrible handling errors from Szczesny.
First the Pole flapped at a cross on the edge of the Arsenal penalty area, which Sagna hacked away as Chadli chased the loose ball. Then, inexplicably, Szczesny failed to collect a high cross into the box, which this time did fall for Chadli, but somehow Koscielny blocked his goalbound shot at point-blank range.
With 20 minutes left, there was a double change for the home side when Paulinho and Gylfi Sigurdsson replaced Chadli and Sandro.
For Arsenal, Rosicky was taken off as defensive midfielder Mathieu Flamini came on.
Wenger was off his seat protesting for what he felt was a penalty after Koscielny was sent tumbling by Jan Vertonghen, but the referee was not interested.
Lloris then made a smart reaction save after Per Mertesacker lashed the ball goalwards from a deep free-kick.
Tempers boiled over in the closing minutes as Sherwood twice hurled the ball at Sagna when Arsenal had been awarded a throw-in.
Arsenal midfielder Mikel Arteta was less than impressed and eventually it needed Wenger to move in and act as peacemaker in a crowded technical area as his side played out the four minutes of stoppage time without further incident.