Germany midfielder Ozil fired home a 71st-minute winner to finally kill off battling Boro, who had dragged themselves back into the game when Alvaro Negredo cancelled out Sanchez's first-half free-kick five minutes after the restart.
The 2-1 triumph - the Gunners' first league away win for three months - lifted Arsene Wenger's men seven points behind fourth-placed Manchester City, with a game in hand.
James Olley was at the Riverside Stadium to assess the key talking points for Arsenal...
Arsenal keep their top four hopes alive
There was no margin for error here after the weekend results and although they were far from convincing, Arsenal edged past a spirited Middlesbrough side to keep their hopes of a top four finish intact.
The Gunners have games in hand on all of the teams above them – except Manchester United – but they will need to improve on this showing if they are to secure a Champions League spot next term.
This was nevertheless a welcome return to winning ways after four straight Premier League losses on the road and will restore a degree of confidence ahead of Sunday’s FA Cup semi-final against Manchester City.
A leopard can change its spots
Arsene Wenger has been heavily criticised for his tactical approach in recent years. Among the main accusations is that nothing ever changes – the same mistakes are made time and again.
It remains to be seen whether 3-4-2-1 is the future for Arsenal, but Wenger’s decision to operate with a back three for the first time since May 1997 was at the very least an experiment worth trying.
It was not an unqualified success here – Middlesbrough were particularly threatening for 20 minutes or so at the start of the second half, either side of Alvaro Negredo’s equaliser – but at least there was a sense of unpredictability about Arsenal after a period of stagnation.
Wenger enjoys a brief respite
The Frenchman cut a nervous, frustrated figure on the touchline for much of this game but the result will quieten the ongoing debate over his future for a few days at least.
Wenger continues to mull over whether he will sign a new contract to stay beyond the end of the season and Arsenal’s top four chances are inextricably linked to his thinking.
It will surely be impossible to sell a two-year deal to the fans as a positive if Arsenal miss out on the Champions League but a win here will at least dampen speculation – that is at least until facing City at Wembley.
Sanchez defies Arsenal’s poor away record
Arsenal have collectively struggled on the road this season but no Premier League player has scored more away goals than Alexis Sanchez after he netted his 13th of the campaign three minutes before the break here.
Sanchez has struggled of late but he produced the one moment of outstanding quality in the first half, underlining why Arsenal have moved to offer him a new contract worth £300,000-a-week.
His body language and attitude have been questioned during Arsenal’s recent slump but he remains the most likely catalyst for a late-season revival.
Ozil answers his critics
Mesut Ozil was not at his most effective but he scored a vital winning goal with a comprehensive finish, which appeared a moment of sweet release after weeks of heavy personal criticism.
Ozil’s 71st-minute strike was his second goal since December 10 and Wenger will hope it is the beginning of a return to form for a player who has become something of a scapegoat when Arsenal under-perform.