Sri Lanka wicketkeeper-batsman Dinesh Chandimal has undergone surgery after injuring his thumb during a domestic match last week.
Walking out with intent and flair, the wicketkeeper-batter smashed a quickfire 51 off just 56 deliveries in the morning session, propelling England to a strong 387.
Spinner Prabath Jayasuriya led the charge with a five-wicket haul as Bangladesh collapsed for just 133 in their second innings, handing Sri Lanka a win by an innings and 78 runs.
Sri Lanka are still 518 runs behind and will need a Herculean effort to avoid the follow-on even though more rain is forecast for Saturday.
The result was the first drawn game in Galle for 12 years.
Australia, who have already secured their place in the final of the World Test Championship against South Africa, won the opening Test, also in Galle, by an innings and 242 runs.
Kusal Mendis' battling half-century salvaged a difficult day one for Sri Lanka as they reached 229/9 in the second Test.
Victory for South Africa followed Test success over West Indies in Guyana in August and Bangladesh in a two-Test series in October.
Smith's unbeaten 120 is his second successive hundred of the series while Carey made a career-best 139 not out hitting 22 fours and three sixes between them.
Sri Lanka's decision to bat first backfired when they were all out for 236 on Day 1 of the first Test.
Sri Lanka put themselves in a good position on the third day of the opening Test against New Zealand after Dimuth Karunaratne and Dinesh Chandimal made half-centuries but the hosts lost some of their momentum to reach 237-4 at stumps in Galle on Friday.
IMAGES from the Day 4 of the 1st Test between England and Sri Lanka in Manchester on Saturday
Dimuth Karunaratne's Test career ended in a whimper as Sri Lanka's most prolific Test opener fell caught behind to Kuhnemann after making 14.
Australia's Usman Khawaja struck a maiden double hundred and Josh Inglis got a century on his Test debut against Sri Lanka to power the tourists to a mammoth 654/6.
Spin twins Matthew Kuhnemann and Nathan Lyon combined to dismantle Sri Lanka and bowl Australia to a crushing victory by an innings and 242 runs in the opening Test in Galle on Saturday.
Steve Smith became the 15th member of Test cricket's 10,000-run club as he and opener Usman Khawaja smashed hundreds to help Australia rack up 330/2 on the opening day of the first Test against Sri Lanka.
Sri Lankan batter Dimuth Karunaratne is set to retire from international cricket after playing his 100th Test -- the second and final Test against Australia, starting in Galle on Thursday.
South Africa require five wickets and Sri Lanka need 143 runs as the second Test heads for a potentially exciting conclusion
Jamie Smith's maiden Test century helped England take a 122-run lead on Day 3 in the first Test against Sri Lanka
South Africa were five wickets from victory as Sri Lanka finished the third day of the first Test on 103-5 in their second innings after being set a 516-run target on Friday.
Nissanka had featured in a 41-run opening partnership with Dimuth Karunaratne, who scored 20 before being undone by a rising delivery from Kagiso Rabada that forced an edge behind to wicketkeeper Kyle Verreynne.
De Silva and veteran Angelo Mathews were not included in the 16-man T20 squad but there was a place for Hasaranga.
New Zealand captain Tim Southee said playing two Tests at Galle, known as a spinner's paradise, was always going to be challenging.
With Sri Lanka's cricket team performing better under Sanath Jayasuriya, he is all set to get a contract extension.
IMAGES from Day 4 of the 2nd Test between England and Sri Lanka, at Lord's, on Sunday.
Kamindu Mendis maintained his red-hot form and struck a majestic 114 to power Sri Lanka to 302/7 on the opening day of the first Test.
Off-spinner Nishan Peiris picked up a six-wicket haul on debut as Sri Lanka crushed New Zealand by an innings and 154 runs.
Sri Lanka took complete control of the second Test against New Zealand on Friday as batting sensation Kamindu Mendis scored the fifth century of his young career and helped the hosts post a mammoth 602-5 declared on day two in Galle.
England captain Ollie Pope returned to form with a classy unbeaten 103 to guide his team to 221/3 on a weather-affected first day of the third and final Test.
India's newly-forged coach-captain combination of Gautam Gambhir and Suryakumar Yadav will look to make instant impact against a Sri Lankan team in disarray.
In-form Sri Lanka right-hander Kamindu Mendis is another player on the move, with the 26-year-old improving five places to 11th and reaching a new career-high rating after he scored another century in the second Test of his side's recent series with New Zealand.
Jamie Smith's unbeaten 72 propelled England to a 23-run lead over Sri Lanka on day two of the opening Test on Thursday
The second match of the series will be played at Lord's, starting from August 29 and the final Test match of the series will take place at Kennington Oval on September 6.
Gus Atkinson scored a superb maiden century as England's lower order collected quick runs against a ragged Sri Lanka bowling attack to post 427
Binura Fernando and Kasun Rajitha, who were named in the original squad, will not tour owing to injuries sustained during the SLC Invitational T20 League 2022.
The fate of the series was decided after India swept aside Sri Lanka in the first two games, leaving the possibility of an embarrassing whitewash hanging over the hosts' heads.
Binura Fernando, Pramod Madushan, Dushan Hemantha back in Sri Lanka squad
Dinesh Chandimal smashed his second half-century of the match to put Sri Lanka in the box seat in the opening Test against Pakistan in Galle on Monday.
Dhananjaya de Silva and Angelo Mathews helped Sri Lanka to overcome a top-order collapse and reach 242-6 on Day one
India captain Rohit Sharma expressed deep concerns about the vitiated air quality in Indian cities.