Media playback is not supported on this device
Only 12.2 overs were possible in two spells, during which Sri Lanka progressed from 32-0 to 78-1 in pursuit of a notional 362 to win.
Kaushal Silva was the sole wicket to fall, lbw to James Anderson, before play was called off at 17:10 BST.
A five-match one-day series starts on 21 June, followed by a Twenty20.
Dimuth Karunaratne finished unbeaten on 37, while Kusal Mendis made 17 not out, hitting Joe Root for six off the final ball of the match.
With both sides taking two points from the draw - Sri Lanka's first in 17 Tests - England lead 10-2 in the inaugural 'Super Series'. There are 12 points available in the limited-overs series.
Media playback is not supported on this device
For all England's frustration at the weather on Monday, they can take credit for the efficient manner in which they completed a 2-0 series win over an inexperienced Sri Lanka side exposed by early summer conditions.
Jonny Bairstow's return of 387 runs was 95 more than the next best batsman and cemented his place in the middle order, while Alex Hales took a sizeable step towards securing the opener's berth by averaging 58.
But England's success was founded on the potency of their attack, led brilliantly by James Anderson with 21 wickets at less than 11 apiece.
Stuart Broad's haul of 12 wickets was bettered only by Anderson, Chris Woakes produced his Test best with bat and ball after replacing the injured Ben Stokes, and Steven Finn hinted at a welcome return to form at Lord's.
England may hold all but one of the nine trophies on offer against countries in bilateral series, but there remain grounds for concern for coach Trevor Bayliss.
Their top-order collapses are still to be eradicated - they have lost their first four wickets before reaching 100 five times in the past seven innings.
The search for a reliable number three goes on after Nick Compton managed only 51 runs in five innings, and it remains to be seen whether James Vince, who scored only 54 runs at 13.5, will be retained at number five for the Pakistan series later this summer.
Despite taking 19 catches - a record for an England wicketkeeper in a three-Test series - Bairstow's glovework has been heavily criticised thanks to three drops and a missed stumping.
England captain Alastair Cook: "It would have been a really good day's cricket. It would have been good to see our guys under some pressure."
Sri Lanka skipper Angelo Mathews: "It would have been an interesting chase but the weather interrupted. We wanted to go for it."
Man of the match and man of the series Jonny Bairstow: "I felt like I had a few people to prove wrong. This series and in South Africa I think I've done that."
Former England batsman Geoffrey Boycott and Test Match Special: "Number three, Compton, and Finn's bowling is up for debate.
"If Stokes isn't fit, Finn will survive. If Stokes is fit, Woakes has out-bowled Finn. Compton has two four-day matches for Middlesex and I don't know which way the selectors will go."

The 21-year-old, from St Austell, was arrested on suspicion of robbery on Thursday and remains in police custody.
A string of robberies and attempted robberies took place at shops in Reawla, Cubert, Scorrier, Porthtowan and Lanner over the last three weeks.
Police said isolated premises were targeted and a knife had been used to threaten staff.
England were denied the chance to press for a series whitewash over Sri Lanka as rain ruined the final day of the third Test at Lord's.
[NEXT_CONCEPT]
A man is being questioned by police over a series of armed robberies which took place across Cornwall.