The 105E was the fourth Ford Anglia, being introduced in 1959. Its American-influenced styling included a sweeping nose line, muted tailfins, and on deluxe versions, a full-width slanted chrome grille in between prominent "eye" headlamps. Basic Anglias featured a narrower, painted grille. Its smoothly sloped bonnet line looked more like a 1950s Studebaker (or even early Ford Thunderbird) than the more aggressive-looking late-'50s American Fords, possibly because its British designers used wind-tunnel testing and streamlining. Its most striking feature was the backward slanting rear window, which was purely a styling gimmick, but it did give good headroom in the back seat.

The new styling was joined by something the smaller Fords had been needing for some time – a new engine – a 997 cc overhead valve (OHV) straight four with an oversquare cylinder bore that became known as the Kent. Acceleration from rest was still sluggish, but it was much improved from earlier cars. Also new for British Fords was a four-speed manual gearbox with synchromesh on the top three forward gears; this was replaced by an all-synchromesh box in September 1962 on 1198 cc powered cars. The notoriously feeble vacuum-powered windscreen wipers of earlier Anglias were replaced with more conventional electrically powered ones.

In 1960, when 191,752 Anglias left Ford's Dagenham plant in the 105E's first full production year, it set a new production-volume record for Ford of Britain. The Anglia Super introduced in September 1962 for the 1963 model year shared the longer-stroke 1198 cc version of the Ford Kent 997 cc engine of the newly introduced Ford Cortina. The Super was distinguished by its painted contrasting-coloured side stripe.

A new Anglia saloon tested by the British Motor magazine in 1959 had a top speed of 73.8 mph (118.8 km/h) and could accelerate from 0–60 mph (97 km/h) in 26.9 seconds. A fuel consumption of 41.2 miles per gallon was recorded. The test car cost £610 including taxes of £180.