LONDON — Three points from the Emirates after Arsenal hung on with 10 men for a 3-2 win over Swansea in the Premier League.

1. Arsenal make it difficult but hang on

Arsenal keep winning, barely, even when they make things extraordinarily difficult. This was a game that showed off the best and worst sides of Arsene Wenger’s team as they turned what should have been a comfortable win into a painfully nervous afternoon. Arsenal led 2-0 and 3-1 and seemed in full control for periods only to have to desperately hang on to a one-goal lead with 10 men for the last 20 minutes.

Granit Xhaka was the main culprit for the Gunners as he first gifted Gylfi Sigurdsson a goal in the first half and then was sent off for a wild lunge at Modou Barrow. The straight red card may have been harsh, but it was a reminder that the Switzerland international’s poor disciplinary record could yet be a costly liability for Wenger’s team. Still, the win maintains a six-game winning streak in the league for Arsenal and gave new Swansea manager Bob Bradley a tough but entertaining introduction to English football.

Arsenal were looking to maintain momentum after a last-gasp winner against Burnley prior to the international break. Saturday’s win puts them level on points with leaders Manchester City after Pep Guardiola’s side mustered a 1-1 draw against Everton.

Theo Walcott bounced back from two subpar performances for England by putting the hosts 2-0 ahead with two quick first-half goals, and Mesut Ozil scored Arsenal’s third with a thumping volley after a perfect ball from Alexis Sanchez. That should have ended Swansea’s resistance, but substitute Gonzalez Borja Baston gave Swansea new life after he was left unmarked in the area. Xhaka was then sent off four minutes later.

Arsenal needed a Petr Cech save to deny Barrow and were lucky to see Sigurdsson blast over the bar as Swansea chased an equaliser that never came.

This was in many ways a crucial test for Arsenal despite the fact that Swansea were looking for their first win since the opening day of the season and playing their first match under a new manager. The Welsh club have been somewhat of a bogey team at the Emirates, collecting three wins and a draw from their last four Premier League visits. The 2-1 win in March was particularly painful as it virtually ended the Gunners’ title hopes and fanned the flames of the fan criticism that Wenger described as a “virus” in the lead-up to this match.

Another slip-up here may have introduced more discontent among supporters but for now, positive vibes remain. Barely.

2. Plenty of positives for Bradley

This was always going to be a tough introduction to the Premier League for Bradley, but even he might be taken aback by the end-to-end action in this game.

Bradley, the first American to take charge of a Premier League club, will likely be livid with the defending on Walcott’s two goals but should also be more than pleased with the way his team responded. Swansea gave Arsenal all they could handle in the second half, with only a bit of luck preventing them from coming away with a result.

Bradley spent much of this game pacing the sideline and shouting instructions at his players, a clear indication that the American is still getting his ideas across to his new squad. But his cautious tactical approach worked well until his team were undone on a quick counterattack midway through the first half. Walcott started and finished off the move after Jordi Amat failed to clear a headed cross from Hector Bellerin. Walcott was again fortunate to see the ball drop to him from a corner for the second goal after Jack Cork failed to connect with a headed clearance.

Such individual mistakes can be worked on in the training ground, but Bradley has clearly instilled a sense of fighting spirit in a squad that must have lacked confidence after their poor run under Francesco Guidolin. That bodes well for their future.

3. Walcott answers his critics

Walcott is making a habit of answering his critics this season. Much has already been made of the way the speedy winger has looked rejuvenated for the Gunners after moving back out to the right flank, but his strong club form only served to highlight two disappointing performances for England during the international break.

The pundits were quickly on his back again after those two games, but Walcott delivered the perfect response with two quick goals here. It’s true that he was helped by lackluster defending, but his goals on Saturday hinted at a player riding a wave of confidence and momentum. His second was especially well-taken, with a quick touch to control the ball before swiveling and firing into the net.

Walcott should arguably have had a hat trick near the end when he came one-on-one with Lukasz Fabianski, but his low shot hit the post. He then inexplicably hit the bar in the final moments after Alex Oxlade-Chamberlain picked him out on a counterattack.

Arsenal’s attack has mainly focused on Sanchez so far this campaign, but the Chile forward looked understandably sluggish after returning late from South America following two grueling World Cup qualifiers. His ball over the top to pick out Ozil for the third goal was sublime, but he looked short of energy and was taken off with 25 minutes to play.

With Olivier Giroud still out injured, Wenger had little choice but to start Sanchez although he may be tempted to give Lucas Perez another outing in Wednesday’s Champions League home match against Ludogorets. Arsenal need Sanchez to keep his form and fitness throughout the season. Wenger must make sure he’s not running on empty before Christmas.