Root, who scored a brilliant first innings 134 against Australia in Cardiff and then 60 in the second, is up from No 7 to No 4 in the new rankings.
He leapfrogged Sri Lanka duo Kumar Sangakkara and Angelo Mathews, plus Younus Khan of Pakistan.
Root now has Hashim Amla firmly within his sights as he is now just seven points behind the South Africa captain after his Cardiff performance has earned him 69 points to give him a career-high 884 points.
The Cardiff Test also saw Australia’s Smith lose the No 1 position to South Africa’s AB de Villiers.
Smith had entered the Test on 913 points but after scores of 33 and 33, he has dropped 12 points to fall to 901 points, seven behind de Villiers.
De Villiers will however miss the two-Test series against Bangladesh, which means a couple of strong performances will put the 26-year-old from Sydney back on top.
Other notable batsmen to head in the right direction following the Cardiff Test include Chris Rogers in 21st (up by two places), Gary Ballance in 31st (up by two places), Ben Stokes in 37th (up by seven places) and Moeen Ali in 65th (up by 19 places).
In the bowling charts, England’s Stuart Broad has claimed fifth position for the first time in his career after rising one place following his five wickets in the match, including three in the second innings.
This has also helped him break the 800-point threshold for the first time as he now trails Pakistan’s Yasir Shah by nine points.
Australia’s Mitchell Johnson – who took just two wickets in Cardiff – has slipped two places to seventh but Mitchell Starc has claimed a career-high 17th position. Starc returned figures of 5-114 and 2- 60, which moved him up three places.
Josh Hazlewood has followed in the footsteps of Starc by also achieving a career-high ranking of 21st. Hazlewood took 5-132, which has propelled him up five places to 21st.
Moeen Ali has followed up his career-high batting ranking with a career-best bowling ranking after his 2-71 and 3-59 has lifted him nine places to 28th spot.
South Africa’s Dale Steyn is the top-ranked bowler, followed by England’s James Anderson and Trent Boult of New Zealand.
Moeen has broken into the top 10 in the Test all-rounders list, rising nine places to No 9. Shakib Al Hasan of Bangladesh remains the No 1.
ICC world Test rankings:
1 AB de Villiers (SA)
2 S Smith (Aus)
3 H Amla (SA)
4 J Root (Eng)
5 K Sangakkara (SL)
6 A Mathews (SL)
7 Y Khan (Pak)
8 K Williamson (NZ)
9 V Kohli (Ind)
10 D Warner (Aus)
1 D Steyn (SA)
2 J Anderson (Eng)
3 T Boult (NZ)
4 Y Shah (Pak)
5 S Broad (Eng)
6 V Philander (SA)
7 M Johnson (Aus)
8 R Herath (SL)
9 M Morkel (SA)
10 T Southee (NZ)