Chelsea claimed a 2-1 victory at Malmo in the Europa League last-32 first leg on Thursday to ease some of the pressure on beleaguered coach Maurizio Sarri, while Arsenal suffered a shock loss to BATE Borisov.
Visitors Chelsea desperately needed a response after back-to-back humiliating away losses, with last Sunday's 6-0 thrashing by Manchester City following a 4-0 reverse at Bournemouth.
Ross Barkley took advantage of a defensive lapse by the home team to put Chelsea ahead on the half-hour mark, with Olivier Giroud adding a 58th-minute second before Malmo's late goal gave them a glimmer of hope ahead of next week's return match at Stamford Bridge.
Malmo did grab the goal their efforts deserved in the 80th minute, as Anders Christiansen slotted past Chelsea goalkeeper Kepa Arrizabalaga.
Chelsea face Manchester United in the FA Cup on Monday, before facing City again in the League Cup final on February 24.
"I am happy because we played with confidence and it wasn't really very easy after the last match (the defeat by City)," Sarri told BT Sport.
"I think I am always confident but sometimes we had the wrong approach with matches so we need to be careful in every match."
Arsenal face an uphill battle to progress after a 1-0 defeat by BATE in a game which saw Alexandre Lacazette sent off for elbowing Aleksandar Filipovic with five minutes remaining.
Stanislav Dragun put BATE ahead just before the break in Barysaw and Arsenal never threatened an equaliser, with Lacazette's late dismissal also ruling him out of next week's second leg in London.
Unai Emery's side have now won just two of their last nine away matches -- last weekend's victory over the Premier League's bottom club Huddersfield and an FA Cup win against third-tier Blackpool.
Elsewhere, Napoli began life without record goalscorer Marek Hamsik with a 3-1 win in a fiery atmosphere at FC Zurich.
Lorenzo Insigne, who has replaced Hamsik as captain after the Slovakian's move to China, scored the opening goal, with Jose Callejon and Piotr Zielinski also on target.
Celtic's European adventures appear to be over for another season after being outclassed 2-0 by Valencia at Parkhead.
Denis Cheryshev and Ruben Sobrino got the goals as the Spaniards extended their recent unbeaten run to six matches.
Inter Milan claimed a 1-0 win at Rapid Vienna despite Mauro Icardi snubbing the trip after being stripped of the captaincy over a contract row, with Lautaro Martinez scoring the winner from the penalty spot.
Wissam Ben Yedder's 22nd-minute strike helped record five-time winners Sevilla take a large step towards the last 16 with a 1-0 victory over Lazio in Rome, while Benfica backed up their 10-0 weekend thumping of Nacional by winning 2-1 at Galatasaray.
Salzburg, semi-finalists last season, lost 2-1 at Club Brugge, while 10-man Sporting Lisbon were beaten 1-0 by Villarreal in the Portuguese capital.