Andre Villas-Boas's team have won all six of their warm-up matches and look in good shape ahead of next Sunday's opener at Stoke.
They conceded for the first time in pre-season when Rangers striker Nikica Jelavic headed the hosts into the lead from a Steven Naismith cross in the sixth minute at Ibrox.
But England Under-21 forward Sturridge - who spent the second half of last season on loan to Bolton - underlined his case for more first-team action at Chelsea by scoring twice before the break.
Substitute Florent Malouda, who had replaced Sturridge on the hour mark, headed in Chelsea's third in the 72nd minute.
The scoreline could have been worse for Rangers -- who crashed out of the Champions League in qualifying round in midweek -- but substitute goalkeeper Neil Alexander saved Frank Lampard's late penalty.
David Goodwillie scored on his Blackburn debut as Steve Kean's side signed off their pre-season preparations with a 4-1 win over Kilmarnock at Rugby Park.
Scotland striker Goodwillie had not even trained with his new colleagues following his move from Dundee United to Rovers on Wednesday, but he was on the scoresheet along with David Dunn and Jason Roberts, who netted twice.
Danny Graham grabbed his first goal for Swansea as Brendan Rodgers' newly-promoted side beat Spanish club Real Betis 1-0 at the Liberty Stadium.
The former Watford striker performed a lovely turn from an Angel Rangel pass to loft the ball into the net in the 63rd minute.
Norwich -- promoted along with Swansea last season -- enjoyed a comfortable 3-0 win over Italian club Parma at Carrow Road.
Canaries captain Grant Holt opened the scoring and winger Andrew Surman added the third as the Canaries warmed up for next weekend's Premier League opener at Wigan.
George Elokobi and Steven Fletcher scored in the space of five minutes midway through the second half to give Wolves a 2-0 win over Real Zaragoza at Molineux.
Sunderland were held to a goalless draw by Hibernian at Easter Road, while Newcastle's match against Fiorentina at St James' Park was abandoned by referee Mark Halsey in the 64th minute because of a waterlogged pitch with the match goalless.
Source: AFP
Source: AFP